%!PS-Adobe-2.0 %%Creator: dvipsk 5.519b Copyright 1986, 1993 Radical Eye Software %%Title: gapspi9.dvi %%Pages: 47 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips gapspi9 %DVIPSSource: TeX output 1995.06.13:1742 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N 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Fn(~)-22 b(r)q Fp(\))20 b(is)f(the)g(spin-indep)q(enden) o(t)g(fermion-fermion)e(in)o(teraction)-13 1148 y(assumed)g(b)q(ounded) h Fn(C)423 1130 y Fi(1)478 1148 y Fp(smo)q(oth)f(and)h(with)g(in)o (teraction)e(range)j Fn(p)1265 1127 y Fi(\000)p Fj(1)1265 1159 y(0)1312 1148 y Fp(.)25 b(If)17 b Fn(\033)h Fp(=)e(0)i(w)o(e)f(sa) o(y)h(that)g(the)-13 1226 y(fermions)f(are)j Fg(spinless)g Fp(while)e(if)h Fn(\033)h Fp(=)f(1)p Fn(=)p Fp(2)h(they)f(are)g Fg(spinning)p Fp(.)31 b(W)l(e)19 b(tak)o(e)g Fn(L)g Fp(=)f Fn(N)5 b(a)p Fp(,)20 b Fn(N)k Fp(in)o(teger.)-13 1305 y(W)l(e)19 b(assume)g(that)h Fn(u)g Fp(and)g Fn(\025)g Fp(are)g Fg(dimensionless)h Fp(and)g Fn(c)p Fp(\()o Fn(~)-23 b(x)o Fp(\))20 b(=)1236 1285 y Fj(\026)-18 b Fl(h)1257 1272 y Ff(2)p 1213 1293 85 2 v 1213 1322 a Fj(2)p Fl(ma)1281 1312 y Ff(2)1304 1305 y Fp(~)-26 b Fn(c)p Fp(\()o Fn(~)j(x=a)p Fp(\),)20 b Fn(v)r Fp(\()n Fn(~)-22 b(r)q Fp(\))20 b(=)1666 1285 y Fj(\026)-18 b Fl(h)1686 1272 y Ff(2)p 1642 1293 V 1642 1322 a Fj(2)p 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b(and)-13 230 y Fn(")p Fp(\(\()p Fm(\000)p Fn(n\031)r(=a)p Fp(\))215 212 y Fi(\000)243 230 y Fn(;)8 b(u)p Fp(\))16 b(is)g(called)f(the)h Fn(n)p Fg(-th)j(ener)n(gy)e(b)n(and)p Fp(.)10 308 y(The)g(zero)f(temp)q (erature)e(Sc)o(h)o(winger)i(functions)g(in)g(a)h(\014nite)e(b)q(o)o(x) i(of)f(size)g Fn(L)g Fp(are)g(de\014ned)g(as)266 424 y(\026)258 436 y Fn(S)r Fp(\()306 423 y Fn(~)309 436 y(k)334 443 y Fj(1)355 436 y Fn(;)8 b(t)395 443 y Fj(1)414 436 y Fn(;)g(")459 443 y Fj(1)478 436 y Fn(;)g(\033)528 443 y Fj(1)547 436 y Fn(;)g(:)g(:)g(:)g(;)655 423 y(~)657 436 y(k)682 443 y Fl(n)706 436 y Fn(;)g(t)746 443 y Fl(n)768 436 y Fn(;)g(")813 443 y Fl(n)837 436 y Fn(;)g(\033)887 443 y Fl(n)910 436 y Fp(\))13 b(=)28 b(lim)994 466 y Fl(L)p Fi(!1)1097 436 y Fm(h)p Fp(\012)p Fm(j)q Fn( )1200 415 y Fl(")1216 420 y Ff(1)1196 450 y Fl(~)1198 458 y(k)1216 463 y Ff(1)1233 458 y Fl(;\033)1263 463 y Ff(1)1293 436 y Fm(\001)258 549 y Fn(e)281 528 y Fj(\()p Fl(t)308 533 y Ff(2)324 528 y Fi(\000)p Fl(t)364 533 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y(is)c(analytic)g(ev)o(erywhere)f (except)h(on)h(t)o(w)o(o)g(p)q(oin)o(ts)g(\(see)f(b)q(elo)o(w\))h(whic) o(h)f(w)o(e)g(call)g Fm(\006)p Fn(p)1607 2509 y Fl(F)1636 2502 y Fp(\(0)p Fn(;)8 b(u;)g(\026)p Fp(\);)23 b(in)-13 2580 y(general)15 b(there)g(could)h(b)q(e)f(more)g(than)h(t)o(w)o(o)f (p)q(oin)o(ts)h(where)g(the)f(o)q(ccupation)h(n)o(um)o(b)q(er)e(is)h (not)i(regular)919 2728 y(3)p eop %%Page: 4 4 4 3 bop -13 -5 a Fp(and)23 b(w)o(e)e(c)o(ho)q(ose)i(among)f(them)f(the) h(one)g(whic)o(h)f(can)i(b)q(e)f(con)o(tin)o(uously)f(parametrized)g(b) o(y)g Fn(\025)i Fp(so)-13 74 y(that)18 b(it)f(reduces)h(to)g Fn(p)405 81 y Fl(F)435 74 y Fp(\(0)p Fn(;)8 b(u;)g(\026)p Fp(\))17 b(if)h Fn(\025)f Fp(=)f(0;)j(if)e(this)h(c)o(hoice)e(is)i(am)o (biguous)f(w)o(e)g(sa)o(y)h(that)g(the)g(F)l(ermi)-13 152 y(momen)o(tum)h(is)k(not)h(w)o(ell)f(de\014ned.)42 b(The)24 b(t)o(w)o(o)f(momen)o(ta)e Fm(\006)p Fn(p)1205 159 y Fl(F)1235 152 y Fp(\()p Fn(\025;)8 b(u;)g(\026)p Fp(\))23 b(are)h(called)e(the)i Fg(F)l(ermi)-13 230 y(surfac)n(e)p Fp(.)10 308 y(An)15 b(imp)q(ortan)o(t)g(role)g(in)g(ph)o(ysics)f(is)i (plaid)f(b)o(y)g(the)g(amplitude)e(of)j(the)f(discon)o(tin)o(uit)o(y)e (at)j(the)f(F)l(ermi)-13 387 y(surface)h Fn(Z)188 368 y Fi(\000)p Fj(1)251 387 y Fp(whic)o(h)g(in)g Fn(d)e Fp(=)g(1)i(is)g(simply)f(de\014ned)h(b)o(y:)752 515 y Fn(Z)789 494 y Fi(\000)p Fj(1)850 515 y Fp(=)d Fn(n)930 528 y Fl(p)948 513 y Fh(\000)948 540 y Fe(F)988 515 y Fm(\000)d Fn(n)1066 528 y Fl(p)1084 513 y Ff(+)1084 540 y Fe(F)1814 515 y Fp(\(7\))10 643 y(Another)16 b(imp)q(ortan)o(t)g(ph)o (ysical)f(quan)o(tit)o(y)g(is:)679 771 y Fn(S)s Fp(\()p Fn(k)r Fp(\))e(=)842 712 y Fk(Z)892 771 y Fn(dte)958 750 y Fi(\000)p Fl(ik)1015 755 y Ff(0)1033 750 y Fl(t)1047 771 y Fn(S)s Fp(\()1096 758 y Fn(~)1099 771 y(k)r(;)8 b(t)p Fp(\))629 b(\(8\))-13 907 y(with)18 b Fn(k)h Fp(=)e(\()215 894 y Fn(~)218 907 y(k)r(;)8 b(k)292 914 y Fj(0)312 907 y Fp(\).)27 b(If)18 b Fm(j)p Fn(\013)p Fm(i)q Fp(,)g Fn(\013)f Fp(=)h(0)p Fn(;)8 b Fp(1)p Fn(;)g(::)p Fp(,)17 b(is)h(a)h(complete)d(set)i(of)h(eigenstates)f(of)g Fn(H)f Fm(\000)12 b Fn(\026N)24 b Fp(and)1827 894 y(~)1816 907 y Fn(E)1852 914 y Fl(\013)-13 985 y Fp(the)16 b(corresp)q(onding)h (eigen)o(v)m(alues,)e(it)h(is)g(easy)g(to)h(c)o(hec)o(k)d(that:)291 1136 y Fn(S)s Fp(\()p Fn(k)r Fp(\))g(=)455 1095 y Fk(X)473 1182 y Fl(\013)621 1090 y Fm(jh)p Fp(\012)p Fm(j)p Fn( )737 1069 y Fi(\000)733 1103 y Fl(~)735 1112 y(k)q(;\033)787 1090 y Fm(j)p Fn(\013)p Fm(i)q(j)866 1072 y Fj(2)p 528 1125 451 2 v 528 1175 a Fm(\000)p Fn(ik)609 1182 y Fj(0)639 1175 y Fp(+)d(\()719 1163 y(~)707 1175 y Fn(E)743 1182 y Fl(\013)779 1175 y Fm(\000)840 1163 y Fp(~)829 1175 y Fn(E)865 1182 y Fj(0)885 1175 y Fp(\)\026)-25 b Fn(h)931 1154 y Fi(\000)p Fj(1)995 1136 y Fp(+)1044 1095 y Fk(X)1063 1187 y Fl(\014)1210 1090 y Fm(jh)p Fp(\012)p Fm(j)q Fn( )1327 1069 y Fj(+)1323 1103 y Fl(~)1325 1112 y(k)q(;\033)1377 1090 y Fm(j)p Fn(\014)s Fm(i)o(j)1454 1072 y Fj(2)p 1117 1125 V 1117 1175 a Fm(\000)p 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Fj(+1)23 1536 y Fl(\013)105 1524 y Fm(\000)11 b Fn(E)196 1506 y Fj(\026)-20 b Fl(n)191 1536 y Fj(0)229 1524 y Fm(\000)10 b Fn(\026)15 b(>)e Fp(0)k(if)f Fn(E)500 1506 y Fj(\026)-20 b Fl(n)p Fj(+1)495 1536 y Fl(\013)583 1524 y Fp(are)16 b(the)g(eigen)o(v)m(alues)f(of)i Fn(H)j Fp(o)o(v)o(er)15 b(the)h(states)h(with)i(\026)-27 b Fn(n)11 b Fp(+)g(1)17 b(particles,)-13 1602 y(and)e Fn(H)t Fm(j)p Fp(\012)p Fm(i)g Fp(=)e Fn(E)299 1584 y Fj(\026)-20 b Fl(n)294 1614 y Fj(0)321 1602 y Fm(j)p Fp(\012)p Fm(i)p Fp(;)15 b(the)g(second)g(addend)g(has)g(p)q(oles)g(at)g Fn(i)p Fp(\026)-25 b Fn(h)q(k)1161 1609 y Fj(0)p Fl(;\014)1227 1602 y Fp(with)14 b Fn(k)1361 1609 y Fj(0)p Fl(;\014)1427 1602 y Fp(=)f Fn(E)1519 1581 y Fj(\026)-20 b Fl(n)q Fi(\000)p Fj(1)1514 1615 y Fl(\014)1594 1602 y Fm(\000)8 b Fn(E)1682 1584 y Fj(\026)-20 b Fl(n)1677 1614 y Fj(0)1711 1602 y Fp(+)8 b Fn(\026)14 b(>)g Fp(0)-13 1680 y(if)i Fn(E)73 1660 y Fj(\026)-20 b Fl(n)p Fi(\000)p Fj(1)68 1693 y Fl(\014)156 1680 y Fp(are)16 b(the)g(eigen)o(v)m(alues)f(of)i Fn(H)j Fp(o)o(v)o(er)c(the)g(states)h(with)h(\026)-26 b Fn(n)11 b Fm(\000)g Fp(1)17 b(particles.)j(The)d Fg(sp)n(e)n(ctr)n (al)f(gap)1844 1668 y Fp(\026)1836 1680 y(\001)-13 1759 y(can)i(b)q(e)h(de\014ned)f(as)i Fn(E)421 1738 y Fj(\026)-20 b Fl(n)p Fj(+1)416 1769 y(0)500 1759 y Fp(+)12 b Fn(E)591 1738 y Fj(\026)-20 b Fl(n)q Fi(\000)p Fj(1)586 1769 y(0)671 1759 y Fm(\000)12 b Fp(2)p Fn(E)787 1740 y Fj(\026)-20 b Fl(n)782 1771 y Fj(0)827 1759 y Fp(where)18 b Fn(E)1011 1738 y Fj(\026)-20 b Fl(n)q Fj(+1)1006 1769 y(0)1078 1759 y Fp(,)19 b Fn(E)1152 1738 y Fj(\026)-20 b Fl(n)p Fi(\000)p Fj(1)1147 1769 y(0)1237 1759 y Fp(are)18 b(the)h(minim)n(um) 14 b(v)m(alues)k(of)h Fn(H)-13 1837 y Fp(o)o(v)o(er)c(the)h(states)g (resp)q(ectiv)o(ely)e(with)k(\026)-26 b Fn(n)11 b Fp(+)f(1,)19 b(\026)-27 b Fn(n)11 b Fm(\000)f Fp(1)17 b(fermions,)d(so)j(that)f(it)g (can)g(b)q(e)g(obtained)g(b)o(y)g(the)-13 1915 y(p)q(oles)g(of)h Fn(S)s Fp(\()p Fn(k)r Fp(\))f(b)o(y)356 1902 y(\026)348 1915 y(\001)d(=)h(min)535 1922 y Fl(\014)567 1915 y Fn(k)592 1922 y Fj(0)p Fl(;\014)655 1915 y Fm(\000)c Fp(min)786 1922 y Fl(\013)819 1915 y Fn(k)844 1922 y Fj(0)p Fl(;\013)896 1915 y Fp(.)10 1993 y(The)16 b(Sc)o(h)o(winger)f(functions)h(of)h(the)e (free)g Fn(\025)g Fp(=)e(0)k(particle)e(system)f(with)i(hamiltonian)e Fn(T)j Fp(+)11 b Fn(uP)23 b Fp(are)-13 2072 y(giv)o(en)15 b(b)o(y)h(the)g(Wic)o(k's)f(rule)g(in)h(term)f(of)h(the)g Fg(pr)n(op)n(agator)f Fn(g)r Fp(\()p Fn(k)r Fp(\),)h(giv)o(en)f(b)o(y)h (eq.\(13\))g(with)g Fn(\025)e Fp(=)g(0:)598 2204 y Fn(g)r Fp(\()p Fn(k)r Fp(\))g(=)997 2170 y(1)p 759 2192 501 2 v 759 2246 a Fm(\000)p Fn(ik)840 2253 y Fj(0)871 2246 y Fp(+)d(\()p Fn(")p Fp(\()978 2233 y Fn(~)981 2246 y(k)q(;)d(u)p Fp(\))j Fm(\000)g Fn(\026)p Fp(\)\026)-25 b Fn(h)1213 2224 y Fi(\000)p Fj(1)1790 2204 y Fp(\(10\))-13 2361 y(F)l(rom)14 b(eq.\(10\))g(it)h(is)g(easy)g(to)g(compute)f(the)h(o)q (ccupation)g(n)o(um)o(b)q(er)f Fn(n)1266 2368 y Fl(~)1268 2376 y(k)1303 2361 y Fp(=)g Fn(\037)p Fp(\()o Fn(")p Fp(\()1443 2348 y Fn(~)1446 2361 y(k)r(;)8 b(u)p Fp(\))g Fm(\000)h Fn(\026)14 b Fm(\024)g Fp(0\))h(where)-13 2439 y Fn(\037)p Fp(\()p Fn(condition)p Fp(\))j(is)g(1)g(if)g(the)g (condition)g(is)g(v)o(eri\014ed)e(and)j(0)f(otherwise.)27 b(The)18 b(F)l(ermi)e(momen)o(tum)e Fn(p)1847 2446 y Fl(F)-13 2517 y Fp(is)19 b(then)h(de\014ned)f(b)o(y)g(the)g(condition)h Fn(")p Fp(\()p Fn(p)767 2524 y Fl(F)796 2517 y Fn(;)8 b(u)p Fp(\))19 b(=)h Fn(\026)p Fp(;)h(note)f(that)g(since)e Fn(")p Fp(\()1388 2504 y Fn(~)1391 2517 y(k)r(;)8 b(u)p Fp(\))19 b(is)h(not)g(con)o(tin)o(uous)-13 2596 y(there)14 b(is)g(not)h(a)g(one)g(to)g(one)g(corresp)q(ondence)f(b)q(et)o(w)o(een) g Fn(p)1064 2603 y Fl(F)1108 2596 y Fp(and)i Fn(\026)p Fp(,)e Fg(i.e.)k Fp(to)d Fn(p)1426 2603 y Fl(F)1470 2596 y Fp(=)e Fn(n\031)r(=a)i Fp(corresp)q(ond)919 2728 y(4)p eop %%Page: 5 5 5 4 bop -13 -5 a Fp(all)18 b(the)h(v)m(alues)g(of)h Fn(\026)f Fp(b)q(elonging)h(to)f([)p Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))888 -23 y Fi(\000)916 -5 y Fn(;)8 b(u)p Fp(\))p Fn(;)g(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1196 -23 y Fj(+)1225 -5 y Fn(;)g(u)p Fp(\)].)28 b(The)19 b(discon)o(tin)o(uit)o(y) e(at)j(the)-13 74 y(F)l(ermi)14 b(surface)i(is)g Fn(Z)376 55 y Fi(\000)p Fj(1)437 74 y Fp(=)e(1.)10 152 y(Unlik)o(e)19 b(to)j(the)f(o)q(ccupation)h(n)o(um)o(b)q(er,)e(the)h(asymptotic)f(b)q (eha)o(viour)h(of)g Fn(g)r Fp(\()p Fn(k)r Fp(\))h(for)f(small)f Fn(k)1757 159 y Fj(0)1798 152 y Fp(and)-13 230 y Fm(j)-2 217 y Fn(~)1 230 y(k)r Fm(j)11 b(\000)f Fn(p)126 237 y Fl(F)172 230 y Fp(dep)q(ends)17 b(on)g(the)f(v)m(alue)g(of)g(the)g(F) l(ermi)e(momen)o(tum)o(:)46 338 y(1.)25 b(if)10 b Fn(\026)i Fp(do)q(es)g(not)g(b)q(elong)g(to)f(the)g(closed)g(set)g([)p Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1073 320 y Fi(\000)1102 338 y Fn(;)d(u)p Fp(\))p Fn(;)g(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1382 320 y Fj(+)1410 338 y Fn(;)g(u)p Fp(\)])i(then)i Fn(g)r Fp(\()p Fn(k)r Fp(\))f(b)q(eha)o(v)o(es)109 416 y(for)17 b(small)e Fn(k)337 423 y Fj(0)374 416 y Fp(and)i Fm(j)480 403 y Fn(~)483 416 y(k)r Fm(j)11 b(\000)g Fn(p)609 423 y Fl(F)656 416 y Fp(as)17 b(\()p Fm(\000)p Fn(ik)816 423 y Fj(0)847 416 y Fp(+)12 b Fn(v)921 423 y Fj(0)940 416 y Fp(\()p Fm(j)970 403 y Fn(~)973 416 y(k)r Fm(j)f(\000)g Fn(p)1099 423 y Fl(F)1129 416 y Fp(\)\))1167 398 y Fi(\000)p Fj(1)1231 416 y Fp(where)16 b Fn(v)1396 423 y Fj(0)1430 416 y Fp(=)1489 397 y Fj(1)p 1488 405 21 2 v 1488 434 a(\026)-18 b Fl(h)1518 393 y(@)r(")p Fj(\()1567 384 y Fl(~)1569 393 y(k)q(;u)p Fj(\))p 1518 405 114 2 v 1555 439 a Fl(@)1574 430 y(~)1576 439 y(k)1637 416 y Fm(j)1649 423 y Fl(~)1651 432 y(k)q Fj(=)p Fl(p)1715 438 y Fe(F)1759 416 y Fg(i.e.)18 b Fp(it)109 495 y(has)d(the)f(same)g(asymptotic)f(b)q (eha)o(viour)h(of)h(the)f(Sc)o(h)o(winger)g(function)g(in)g(the)g Fn(u)g Fp(=)g(0)h(case)f(\(see)109 573 y(eq.\(12\))i(b)q(elo)o(w\).)46 689 y(2.)25 b(quite)14 b(di\013eren)o(t)h(is)g(the)h(case)f(in)g(whic)o (h)g Fn(\026)f Fm(2)g Fp([)p Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1145 671 y Fi(\000)1174 689 y Fn(;)8 b(u)p Fp(\))p Fn(;)g(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1454 671 y Fj(+)1482 689 y Fn(;)g(u)p Fp(\)])14 b Fg(i.e.)k Fn(p)1688 696 y Fl(F)1732 689 y Fp(=)c Fn(\031)r(=a)p Fp(.)109 767 y(In)i(this)g(case)g(w)o(e)g(ha)o(v)o(e)f(that:)727 882 y(lim)619 921 y Fi(j)627 912 y Fl(~)629 921 y(k)q Fi(j!)p Fj(\()p Fl(\031)q(=a)p Fj(\))779 912 y Fh(\006)804 921 y Fl(;k)832 926 y Ff(0)850 921 y Fi(!)p Fj(0)911 882 y Fn(g)r Fp(\()p Fn(k)r Fp(\))f(=)1207 848 y(\026)-25 b Fn(h)p 1072 870 298 2 v 1072 916 a(")p Fp(\(\()p Fn(\031)r(=a)p Fp(\))1232 901 y Fi(\006)1261 916 y Fp(\))11 b Fm(\000)f Fn(\026)1790 882 y Fp(\(11\))10 1031 y(The)24 b(ab)q(o)o(v)o(e)f (limits)e(are)i(\014nite)g(unless)g(one)g(c)o(ho)q(oses)h Fn(\026)i Fp(=)g Fn(")p Fp(\(\()p Fn(\031)r(=a)p Fp(\))1333 1013 y Fi(\000)1362 1031 y Fp(\))d(or)h Fn(\026)i Fp(=)g Fn(")p Fp(\(\()p Fn(\031)r(=a)p Fp(\))1750 1013 y Fj(+)1778 1031 y Fp(\),)f(in)-13 1109 y(whic)o(h)19 b(cases)h(one)g(of)g(the)g(t) o(w)o(o)g(limits)d(is)j(singular.)32 b(Of)20 b(course)f(the)h(ph)o (ysical)f(prop)q(erties)h(do)g(not)-13 1187 y(dep)q(end)c(on)g(the)g(c) o(hoice)f(of)h Fn(\026)e Fm(2)g Fp([)p Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))800 1169 y Fi(\000)829 1187 y Fn(;)8 b(u)p Fp(\))p Fn(;)g(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1109 1169 y Fj(+)1137 1187 y Fn(;)g(u)p Fp(\)])15 b(:)21 b(for)16 b(instance)g(the)g(sp)q(ectral)g(gap)-13 1266 y(de\014ned)i(ab)q(o)o(v) o(e)h(in)f(the)g Fn(\025)h Fp(=)e(0)i(case)g(is)f(equal)g(to)h Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1127 1247 y Fj(+)1156 1266 y Fn(;)8 b(u)p Fp(\))k Fm(\000)h Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1478 1247 y Fi(\000)1506 1266 y Fn(;)8 b(u)p Fp(\))19 b(if)f Fn(p)1665 1273 y Fl(F)1712 1266 y Fp(=)g Fn(n\031)r(=a)-13 1344 y Fp(for)e(an)o(y)f(c)o(hoice)g (of)h Fn(\026)g Fp(and)g(it)f(is)h(0)g(for)g(all)f(the)h(other)f(v)m (alues)h(of)g Fn(p)1203 1351 y Fl(F)1233 1344 y Fp(.)21 b(It)15 b(is)h(con)o(v)o(enien)o(t)d(to)j(de\014ne)g(the)-13 1422 y(adimensional)f(sp)q(ectral)i(gap,)h(if)e Fn(p)640 1429 y Fl(F)685 1422 y Fp(=)f Fn(\031)r(=a)p Fp(,)h(as)i(\001)950 1429 y Fl(n)973 1422 y Fp(\()p Fn(u)p Fp(\))d(=)1111 1402 y Fj(2)p Fl(ma)1179 1391 y Ff(2)p 1111 1410 85 2 v 1135 1441 a Fj(\026)-18 b Fl(h)1155 1428 y Ff(2)1201 1422 y Fp([)p Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1404 1404 y Fj(+)1433 1422 y Fn(;)8 b(u)p Fp(\))j Fm(\000)g Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))1752 1404 y Fi(\000)1781 1422 y Fn(;)d(u)p Fp(\)].)-13 1500 y(F)l(or)18 b(small)f Fn(u)g Fp(w)o(e)h(ha)o(v)o(e)g(\(see)f(App)q(endix)h(2\),)g(\001)881 1507 y Fj(1)901 1500 y Fp(\()p Fn(u)p Fp(\))e Fm(\021)h Fp(\001\()p Fn(u)p Fp(\))f(=)h Fn(c)1238 1507 y Fj(1)1258 1500 y Fn(u)12 b Fp(+)h Fn(O)q Fp(\()p Fn(u)1434 1482 y Fj(2)1454 1500 y Fp(\))18 b(where)g Fn(c)1655 1507 y Fj(1)1674 1500 y Fp(,)g(the)g(\014rst)-13 1579 y(F)l(ourier)d(co)q (e\016cien)o(t)g(of)j(~)-25 b Fn(c)p Fp(\()o Fn(~)i(x)o Fp(\),)16 b(is)g(assumed)g(from)f(no)o(w)i(on)f(equal)g(to)h(1.)10 1657 y(If)f Fn(u)e Fm(\021)f Fp(0)k(the)f(propagator)i(eq.\(10\))e(is)g (giv)o(en)g(b)o(y:)605 1775 y Fn(g)628 1782 y Fj(free)686 1775 y Fp(\()p Fn(k)r Fp(\))e(=)1025 1742 y(1)p 822 1764 432 2 v 822 1817 a Fm(\000)p Fn(ik)903 1824 y Fj(0)933 1817 y Fp(+)987 1798 y Fj(\026)-18 b Fl(h)1008 1785 y Fh(\000)p Ff(1)p 987 1806 62 2 v 994 1835 a Fj(2)p Fl(m)1054 1817 y Fp(\()1070 1804 y Fn(~)1073 1817 y(k)1100 1803 y Fj(2)1131 1817 y Fm(\000)11 b Fn(p)1205 1800 y Fj(2)1205 1830 y Fl(F)1234 1817 y Fp(\))1790 1775 y(\(12\))-13 1914 y(whic)o(h)k(is)g(singular)h(for)g(an)o(y)g(v)m(alue)f(of)h(the)g (c)o(hemical)c(p)q(oten)o(tial)j Fn(\026)g Fp(=)e Fn(p)1310 1896 y Fj(2)1310 1927 y Fl(F)1340 1914 y Fn(=)p Fp(2)p Fn(m)p Fp(;)j(moreo)o(v)o(er)e(the)h(F)l(ermi)-13 1993 y(momen)o(tum)e(is)k(giv)o(en)f(b)o(y)g Fn(\026)g Fp(=)593 1966 y Fj(\026)-18 b Fl(h)613 1953 y Ff(2)631 1966 y Fl(p)649 1954 y Ff(2)649 1978 y Fe(F)p 593 1981 81 2 v 609 2010 a Fj(2)p Fl(m)679 1993 y Fp(,)17 b(the)g(sp)q(ectral)g(gaps) h(are)f(zero)g(and)g(the)g(discon)o(tin)o(uit)o(y)e(at)j(the)-13 2071 y(F)l(ermi)c(surface)i Fn(Z)327 2053 y Fi(\000)p Fj(1)388 2071 y Fp(=)e(1.)10 2149 y(The)j(free)e(Sc)o(h)o(winger)g (function)i Fn(S)656 2156 y Fj(0)675 2149 y Fp(\()p Fn(x;)8 b(y)r Fp(\),)15 b(obtained)i(replacing)e(eq.\(10\))h(in)g(eq.\(5\),)f Fg(i.e.)200 2268 y Fn(S)230 2275 y Fj(0)250 2268 y Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)428 2209 y Fk(Z)478 2268 y Fn(dk)r(\036)p Fp(\()575 2255 y Fn(~)578 2268 y(k)s(;)7 b(~)-23 b(x)o(;)8 b(u)p Fp(\))p Fn(\036)p Fp(\()769 2255 y Fn(~)772 2268 y(k)r(;)g Fm(\000)o Fn(~)-23 b(y)q(;)8 b(u)p Fp(\))p Fn(e)977 2247 y Fl(ik)1007 2252 y Ff(0)1024 2247 y Fj(\()p Fl(x)1058 2252 y Ff(0)1074 2247 y Fi(\000)p Fl(y)1118 2252 y Ff(0)1137 2247 y Fj(\))1396 2234 y Fp(1)p 1157 2256 501 2 v 1157 2310 a Fm(\000)p Fn(ik)1238 2317 y Fj(0)1269 2310 y Fp(+)j(\()p Fn(")p Fp(\()1376 2297 y Fn(~)1379 2310 y(k)r(;)d(u)p Fp(\))i Fm(\000)h Fn(\026)p Fp(\)\026)-25 b Fn(h)1611 2288 y Fi(\000)p Fj(1)1790 2268 y Fp(\(13\))-13 2417 y(deca)o(y)15 b(for)i(large)f(distance)g Fm(j)p Fn(x)11 b Fm(\000)f Fn(y)r Fm(j)k Fp(=)713 2366 y Fk(q)p 755 2366 487 2 v 51 x Fp(\()o Fn(~)-23 b(x)11 b Fm(\000)f Fn(~)-23 b(y)q Fp(\))907 2402 y Fj(2)938 2417 y Fp(+)11 b Fn(v)1013 2399 y Fj(2)1011 2427 y(0)1032 2417 y Fp(\()p Fn(x)1079 2424 y Fj(0)1109 2417 y Fm(\000)g Fn(y)1183 2424 y Fj(0)1203 2417 y Fp(\))1222 2402 y Fj(2)1258 2417 y Fp(:)46 2525 y(1.)25 b(if)19 b Fn(u)g Fp(=)g(0)h(or)g Fn(u)f Fm(6)p Fp(=)g(0)h(but)g Fn(p)633 2532 y Fl(F)682 2525 y Fm(6)p Fp(=)f Fn(n\031)r(=a)p Fp(,)g(as)i Fn(f)5 b Fp(\()o Fn(~)-23 b(x;)8 b(~)-24 b(y)q Fp(\))p Fn(=)p Fm(j)p Fn(x)13 b Fm(\000)g Fn(y)r Fm(j)p Fp(,)19 b(where)h Fn(f)5 b Fp(\()o Fn(~)-23 b(x;)8 b(~)-24 b(y)q Fp(\))20 b(is)f(a)h(b)q(ounded)109 2603 y(function)c(equal)f(to)i(sin\()p Fn(p)591 2610 y Fl(F)621 2603 y Fp(\()o Fn(~)-23 b(x)10 b Fm(\000)h Fn(~)-24 b(y)r Fp(\)\))16 b(if)g Fn(u)d Fp(=)h(0;)919 2728 y(5)p eop %%Page: 6 6 6 5 bop 46 -5 a Fp(2.)25 b(if)15 b Fn(u)f Fm(6)p Fp(=)g(0)i(and)h Fn(p)406 2 y Fl(F)450 -5 y Fp(=)c Fn(n\031)r(=a)j Fp(deca)o(ys)g (faster)g(than)h(an)o(y)f(p)q(o)o(w)o(er)g(for)h(large)f(distances)g Fg(i.e.)i Fp(for)e(all)109 74 y Fn(N)21 b Fp(one)c(can)f(\014nd)h(a)f (constan)o(t)h Fn(C)718 81 y Fl(N)768 74 y Fp(suc)o(h)f(that)692 189 y Fm(j)p Fn(S)736 196 y Fj(0)756 189 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fm(j)k(\024)1018 155 y Fn(C)1053 162 y Fl(N)1087 155 y Fp(\001)1128 162 y Fl(n)1151 155 y Fn(a)1177 137 y Fi(\000)p Fj(1)p 954 177 335 2 v 954 223 a Fp(1)f(+)g(\001)1079 208 y Fl(N)1079 235 y(n)1113 223 y Fm(j)p Fn(x)f Fm(\000)h Fn(y)r Fm(j)1255 208 y Fl(N)1790 189 y Fp(\(14\))109 309 y(\(note)16 b(that)h Fn(S)372 316 y Fj(0)391 309 y Fp(\()p Fn(x;)8 b(y)r Fp(\))16 b(has)h(the)f(dimension)e(of)j(an)g (in)o(v)o(erse)d(length,)p Fg(i.e.)j Fp(of)g Fn(a)1520 291 y Fi(\000)p Fj(1)1567 309 y Fp(\).)10 412 y(The)e(free)f Fn(\025)g Fp(=)g(0)h(system)e(is)i(called)f(a)h Fg(metal)g Fp(if)f Fn(p)915 419 y Fl(F)959 412 y Fm(6)p Fp(=)1016 392 y Fl(n\031)p 1016 400 43 2 v 1028 429 a(a)1078 412 y Fp(while)g(if)g Fn(p)1271 419 y Fl(F)1315 412 y Fp(=)1372 392 y Fl(n\031)p 1372 400 V 1384 429 a(a)1434 412 y Fp(and)h Fn(u)f Fm(6)p Fp(=)g(0)h(it)f(is)h(called)-13 490 y(an)h Fg(insulator)p Fp(:)21 b(w)o(e)16 b(can)g(extend)f(to)h(the)f Fn(\025)f Fm(6)p Fp(=)g(0)i(case)g(suc)o(h)f(de\014nitions)h(b)o(y)f (sa)o(ying)h(that)g(a)g(fermionic)-13 569 y(system)i(in)i(whic)o(h)f(b) q(oth)h Fn(S)s Fp(\(0)p Fn(;)8 b(p)596 548 y Fj(+)596 581 y Fl(F)626 569 y Fp(\))20 b(and)g Fn(S)s Fp(\(0)p Fn(;)8 b(p)885 548 y Fi(\000)885 581 y Fl(F)916 569 y Fp(\))19 b(are)h(in\014nite)f(is)g(a)i(metal)d(while)g(if)i(at)g(least) f(one)-13 647 y(b)q(et)o(w)o(een)c Fn(S)s Fp(\(0)p Fn(;)8 b(p)297 626 y Fj(+)297 659 y Fl(F)327 647 y Fp(\))16 b(or)h Fn(S)s Fp(\(0)p Fn(;)8 b(p)544 626 y Fi(\000)544 659 y Fl(F)574 647 y Fp(\))16 b(is)g(\014nite)g(it)g(is)g(an)h (insulator.)10 725 y(W)l(e)g(consider)g(no)o(w)h(what)g(happ)q(ens)h (in)d(presence)h(of)h(an)f(in)o(teraction)g Fn(\025)f Fm(6)p Fp(=)f(0.)25 b(W)l(e)17 b(note)h(\014rst)f(that)-13 803 y(generally)h Fn(p)221 810 y Fl(F)251 803 y Fp(\()p Fn(\025;)8 b(u;)g(\026)p Fp(\))18 b Fm(6)p Fp(=)h Fn(p)517 810 y Fl(F)547 803 y Fp(\(0)p Fn(;)8 b(u;)g(\026)p Fp(\),)19 b Fg(i.e.)f Fp(the)h(in)o(teraction)f(c)o(hanges)h(the)g(F)l(ermi)d (momen)o(tum)f(at)-13 882 y(\014xed)h Fn(\026)p Fp(.)10 960 y(Since)g(the)h(early)f(w)o(orks)h(on)g(the)f(theory)h(of)g(F)l (ermi)d(systems,)h(\(L.W.\),)h(it)h(has)g(b)q(een)g(realized)e(that)-13 1038 y(it)f(is)h(more)e(natural)j(to)f(study)g(the)g(prop)q(erties)f (of)i(w)o(eakly)d(in)o(teracting)h(fermionic)f(systems)g(when)i Fn(\025)-13 1116 y Fp(is)g(v)m(aried)f(at)i(\014xed)f(F)l(ermi)d(momen) o(tum)f(rather)k(than)h(at)f(\014xed)g(c)o(hemical)d(p)q(oten)o(tial.) 20 b(Therefore)15 b(w)o(e)-13 1194 y(write)g(the)g(c)o(hemical)e(p)q (oten)o(tial)i(in)g(eq.\(1\))g(as)h Fn(\026)f Fp(=)e Fn(\026)976 1201 y Fj(0)1006 1194 y Fp(+)d Fn(\027)19 b Fp(where)c Fn(\026)1266 1201 y Fj(0)1302 1194 y Fp(is)g(the)g(c)o (hemical)e(p)q(oten)o(tial)i(of)-13 1273 y(a)h(fermionic)e(system)h (with)h(F)l(ermi)e(momen)o(tum)e Fn(p)932 1280 y Fl(F)978 1273 y Fp(and)17 b(hamiltonian)e Fn(T)i Fp(+)11 b Fn(uP)c Fp(.)10 1351 y(In)16 b(the)g(case)h Fn(p)283 1358 y Fl(F)326 1351 y Fp(=)d Fn(\031)r(=a)i(\026)503 1358 y Fj(0)539 1351 y Fp(is)g(c)o(hosen)g(at)h(the)f(cen)o(ter)f(of)h(the)g(in)o(terv) m(al)-13 1429 y([)p Fn(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))190 1411 y Fi(\000)218 1429 y Fn(;)8 b(u)p Fp(\))p Fn(;)g(")p Fp(\(\()p Fn(n\031)r(=a)p Fp(\))498 1411 y Fj(+)526 1429 y Fn(;)g(u)p Fp(\)];)14 b(suc)o(h)h(c)o(hoice)e(is)i (rather)g(arbitrary)l(,)f(as)i(one)e(could)h(tak)o(e)f(an)o(y)h(v)m (alue)-13 1507 y(in)22 b(that)g(in)o(terv)m(al)f(and)i(w)o(e)f(mak)o(e) e(it)i(for)g(de\014niteness.)38 b(W)l(e)22 b(\014x)g(the)g(\\coun)o (terterm")f Fn(\027)k Fp(so)e(that)-13 1586 y Fn(p)11 1593 y Fl(F)41 1586 y Fp(\()p Fn(\025;)8 b(u;)g(\026)189 1593 y Fj(0)209 1586 y Fp(+)p Fn(\027)s Fp(\))14 b(=)g Fn(p)383 1593 y Fl(F)412 1586 y Fp(\(0)p Fn(;)8 b(u;)g(\026)556 1593 y Fj(0)576 1586 y Fp(\))14 b(=)g Fn(p)685 1593 y Fl(F)715 1586 y Fp(,)d Fg(i.e.)18 b(we)13 b(for)n(c)n(e)f(the)h(F)l (ermi)f(momentum)i(in)e(the)h(\\inter)n(acting")-13 1664 y Fn(\025)i Fm(6)p Fp(=)g(0)k Fg(the)n(ory)e(and)h(\\fr)n(e)n(e")f Fn(\025)f Fp(=)e(0)19 b Fg(the)n(ory)e(to)h(b)n(e)h(the)f(same)p Fp(.)23 b(According)17 b(to)g(the)g(formal)e(Luttinger)-13 1742 y(theorem)e(\(L1.\),)h(\(B.G.L.\),)f(\014xing)i Fn(p)676 1749 y Fl(F)705 1742 y Fp(\()p Fn(\025;)8 b(u;)g(\026)853 1749 y Fj(0)881 1742 y Fp(+)f Fn(\027)s Fp(\))14 b(should)h(b)q(e)g (equiv)m(alen)o(t)d(to)j(\014xing)f(the)h(ph)o(ysical)-13 1820 y(densit)o(y)f Fn(\032)p Fp(\()p Fn(\025;)8 b(u;)g(\026)327 1827 y Fj(0)355 1820 y Fp(+)h Fn(\027)s Fp(\),)15 b Fg(i.e.)j Fn(p)586 1827 y Fl(F)615 1820 y Fp(\()p Fn(\025;)8 b(u;)g(\026)763 1827 y Fj(0)792 1820 y Fp(+)g 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(action)g Fn(\025)g Fg(is)g(smal)r(l)h(with)f(r)n(esp)n(e)n(ct)f(to)h (the)g(gap)g(amplitude)g Fp(\001\()p Fn(u)p Fp(\))f Fg(of)h(the)g(fr)n (e)n(e)919 2728 y Fp(6)p eop %%Page: 7 7 7 6 bop -13 -5 a Fg(system)16 b Fp(one)g(exp)q(ects)g(from)f(ph)o (ysical)g(argumen)o(ts)g(that)i(the)f(b)q(eha)o(viour)g(of)g(the)g (free)g(system)e(is)i(not)-13 74 y(di\013eren)o(t)f(from)g(the)h(b)q (eha)o(viour)h(in)f(presence)f(of)h(small)f(in)o(teractions.)10 152 y(This)26 b(is)g(in)f(fact)h(what)g(w)o(e)f(\014nd)h(\(see)f(the)h (\014rst)g(statemen)o(t)e(of)i(the)f(theorem)f(b)q(elo)o(w\))i(as)g(w)o (e)-13 230 y(obtain)f(that)f(the)g(Sc)o(h)o(winger)f(function)h(is)g (giv)o(en)g(b)o(y)g Fn(S)1088 237 y Fj(0)1107 230 y Fp(\()p Fn(x;)8 b(y)r Fp(\))16 b(+)g(\()p Fn(\025=)p Fp(\001\()p Fn(u)p Fp(\)\))1497 217 y(~)1488 230 y Fn(S)s Fp(\()p Fn(x;)8 b(y)r Fp(\))24 b(with)g(b)q(oth)-13 308 y Fn(S)17 315 y Fj(0)37 308 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fn(;)180 296 y Fp(~)173 308 y Fn(S)q Fp(\()p Fn(x;)g(y)r Fp(\))14 b(ob)q(eying)h(to)f(the)h(b)q(ound)g(eq.\(14\).)20 b(W)l(e)15 b(\014x)f(the)g(coun)o(terterm)e Fn(\027)17 b Fp(=)d(0)h Fg(but)i(fr)n(om)d(the)-13 387 y(pr)n(o)n(of)h(it)h(wil)r(l)i(app)n(e)n (ar)d(that)i(we)g(c)n(ould)g(e)n(qual)r(ly)h(\014x)g Fn(\027)f Fp(=)c Fn(O)q Fp(\()p Fn(\025)p Fp(\);)k(this)e(means)f(that) i(in)f(this)g(in)o(teracting)-13 465 y(system)f(as)i(in)f(the)g(free)f (one,)h(in)g(the)g(\014lled)g(band)h(case,)f(the)g(c)o(hemical)d(p)q (oten)o(tial)j Fg(is)h(not)h(c)n(ompletely)-13 543 y(determine)n(d)g Fp(b)o(y)e(the)h(F)l(ermi)e(momen)o(tum)o(.)10 621 y(More)f(in)o (teresting)f(is)h(the)g(case)g(in)g(whic)o(h)g Fg(the)i(str)n(ength)g (of)f(the)h(inter)n(action)h Fn(\025)f Fg(is)f(lar)n(ge)h(with)g(r)n (esp)n(e)n(ct)-13 700 y(to)21 b(the)g(gap)h(amplitude)f Fp(\001\()p Fn(u)p Fp(\))g Fg(of)f(the)i(fr)n(e)n(e)e(system)p Fp(;)i(in)e(this)g(case)g(in)f(fact)h(one)h(exp)q(ects)e(that)i(the)-13 778 y(in)o(teraction)g(could)h(c)o(hange)g(in)g(a)g(relev)m(an)o(t)f(w) 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(teracting)-13 2577 y(fermions)i(is)h(called)g(a)h Fg(normal)g(liquid)h Fp(if)e(the)g(o)q(ccupation)h(n)o(um)o(b)q(er)e(discon)o(tin)o(uit)o(y) f(is)j Fn(O)q Fp(\(1\),)g(while)919 2728 y(7)p eop %%Page: 8 8 8 7 bop -13 -5 a Fp(it)19 b(is)h(called)e(a)j Fg(Luttinger)g(liquid)g Fp(if,)e(for)h(small)f Fm(j)914 -18 y Fn(~)917 -5 y(k)q Fm(j)p Fp(,)i Fn(n)1021 11 y Fl(p)1039 17 y Fe(F)1064 11 y Fj(+)1089 2 y Fl(~)1091 11 y(k)1126 -5 y Fm(\000)13 b Fn(n)1207 11 y Fl(p)1225 17 y Fe(F)1251 11 y Fi(\000)1276 2 y Fl(~)1278 11 y(k)1319 -5 y Fm(')20 b(j)1389 -18 y Fn(~)1392 -5 y(k)r Fm(j)1433 -23 y Fl(\021)1473 -5 y Fp(with)g Fn(\021)i Fp(=)d Fn(O)q Fp(\()p Fn(\025)1776 -23 y Fj(2)1797 -5 y Fp(\))h(in)-13 74 y(general.)28 b(In)19 b(analogy)g(with)g(the)f(ab)q(o)o(v)o(e)h(de\014nition)f(in)g (the)h(case)g(of)g(\014lled)e(band)j(w)o(e)e(can)h(de\014ne)f(a)-13 152 y Fg(normal)f(insulator)g Fp(a)g(F)l(ermi)d(system)h(in)h(whic)o(h) g(for)g(all)g Fn(u)h Fp(it)e(is)i Fn(Z)1219 134 y Fi(\000)p Fj(1)1280 152 y Fp(=)d Fn(O)q Fp(\(1\),)j(while)e(w)o(e)h(can)h (de\014ne)-13 230 y(a)j Fg(Luttinger)i(insulator)e Fp(a)h(system)d(in)i (whic)o(h)f(the)h(quan)o(tit)o(y)e Fn(Z)1193 212 y Fi(\000)p Fj(1)1260 230 y Fp(v)m(anishes)j(as)f Fn(u)g Fm(!)g Fp(0)h(\(whic)o(h)e (is)-13 308 y(therefore)i(an)h Fg(anomalous)h(b)n(ehaviour)p Fp(\).)38 b(W)l(e)22 b(can)g(then)g(conclude)f(that)h(our)h(\014lled)d (band)j(F)l(ermi)-13 387 y(system)15 b(is)h(a)g(Luttinger)h(insulator.) 10 465 y(W)l(e)12 b(are)h(not)f(really)f(able)h(to)h(compute)e(the)h (full)f(analytic)h(structure)g(near)g(the)g(F)l(ermi)e(surface)i(in)g (the)-13 543 y(complex)g Fn(k)r Fp(-plane)j(\(hence)g(the)f(p)q(oles\)) h(of)h Fn(S)s Fp(\()p Fn(k)r Fp(\))e(so)i(that)f(w)o(e)g(do)g(not)g (really)f(kno)o(w)h(the)g(sp)q(ectral)g(gap)-13 621 y(in)g(the)g(in)o (teracting)g(case;)g(ho)o(w)o(ev)o(er)f(our)i(results)g(pro)o(vide)f (in)g(our)h(opinion)f(a)h(v)o(ery)f(strong)h(evidence)-13 700 y(that)d(there)g(is)g(a)h(gap)g(of)f(order)h Fn(O)q Fp(\()p Fn(u)646 681 y Fj(1)p Fi(\000)p Fl(\021)709 686 y Ff(2)728 700 y Fp(\))f Fg(i.e.)18 b Fp(that)c(the)e(in)o(teraction)h (c)o(hanges)g(the)g(amplitude)e(of)j(the)-13 778 y(gap)h(so)g(that)g (the)f(ratio)h(b)q(et)o(w)o(een)e(the)h(free)g(and)h(the)f(in)o (teracting)f(gap)i(is)g Fn(O)q Fp(\()p Fn(u)1437 760 y Fi(\000)p Fl(\021)1482 765 y Ff(2)1501 778 y Fp(\))g Fg(i.e.)i Fp(v)m(anishing)e(or)-13 856 y(div)o(erging)e(as)i Fn(u)f Fm(!)f Fp(0)i(dep)q(ending)f(on)h(the)f(repulsiv)o(e)f(or)h (attractiv)o(e)f(nature)i(of)f(the)g(in)o(teraction.)20 b(W)l(e)-13 934 y(think)e(that)h(a)g(more)f(careful)g(study)h(of)g(the) g(analytic)f(structure)g(of)h Fn(S)s Fp(\()p Fn(k)r Fp(\))g(will)f (allo)o(w)g(us)h(to)g(pro)o(v)o(e)-13 1013 y(the)d(ab)q(o)o(v)o(e)g (claims)e(ab)q(out)k(the)e(gap.)10 1091 y(An)j(analogous)j(statemen)o (t)17 b(w)o(as)j(claimed)d(in)i(a)h(similar)d(one)j(dimensional)d (fermionic)g(mo)q(del)h(b)o(y)-13 1169 y(\(L.E.1\))d(on)g(the)f(basis)h (of)g(alternativ)o(e)f(heuristic)f(argumen)o(ts)h(\(and)h(it)f(seems)f (quite)h(di\016cult)f(to)i(put)-13 1247 y(them)f(in)i(a)h(rigourous)g (form,)e(see)h(\(S.\)\).)10 1326 y(Our)k(results)f(are)h(summarized)d (b)o(y)i(the)g(follo)o(wing)h(theorem)e(whic)o(h)g(is)i(form)o(ulated)e (in)h(terms)f(of)-13 1404 y(the)e(follo)o(wing)g(quan)o(tities:)46 1519 y(1.)25 b Fn(\026)138 1526 y Fj(0)172 1519 y Fp(=)13 b(\()p Fn(")p Fp(\()p Fn(\031)r(=a)p Fp(\))383 1501 y Fi(\000)412 1519 y Fn(;)8 b(u)p Fp(\))j(+)g Fn(")p Fp(\(\()p Fn(\031)r(=a)p Fp(\))701 1501 y Fj(+)729 1519 y Fn(;)d(u)p Fp(\))p Fn(=)p Fp(2:)22 b(`)p Fg(unp)n(erturb)n(e)n(d)17 b(chemic)n(al)h(p)n(otential)46 1638 y Fp(2.)25 b Fn(\026)14 b Fp(=)g Fn(\026)233 1645 y Fj(0)264 1638 y Fp(+)d Fn(\027)s Fp(:)21 b Fg(chemic)n(al)e(p)n(otential)46 1756 y Fp(3.)25 b(\001\()p Fn(u)p Fp(\))13 b(=)286 1737 y Fj(2)p Fl(ma)354 1725 y Ff(2)p 286 1744 85 2 v 309 1775 a Fj(\026)-18 b Fl(h)330 1762 y Ff(2)376 1756 y Fp([)p Fn(")p Fp(\()p Fn(\031)r(=a)p Fp(\))531 1738 y Fj(+)559 1756 y Fn(;)8 b(u)p Fp(\))i Fm(\000)h Fn(")p Fp(\(\()p Fn(\031)r(=a)p Fp(\))848 1738 y Fi(\000)877 1756 y Fn(;)d(u)p Fp(\)]:)20 b Fg(adimensional)f(unp)n(erturb)n(e)n(d)e(gap)46 1874 y Fp(4.)25 b Fn(v)133 1881 y Fj(0)166 1874 y Fp(=)14 b(\026)-25 b Fn(h\031)r(=ma)p Fp(:)21 b Fg(F)l(ermi)c(velo)n(city)46 1993 y Fp(5.)25 b Fm(j)p Fn(k)r Fm(j)13 b Fp(=)h Fn(a)255 1940 y Fk(q)p 296 1940 385 2 v 296 1993 a Fn(v)322 1972 y Fi(\000)p Fj(2)320 2004 y(0)369 1993 y Fn(k)396 1976 y Fj(2)394 2004 y(0)427 1993 y Fp(+)d(\()p Fm(j)506 1980 y Fn(~)509 1993 y(k)q Fm(j)g(\000)615 1973 y Fl(\031)p 615 1981 22 2 v 616 2010 a(a)642 1993 y Fp(\))661 1978 y Fj(2)680 1993 y Fp(:)21 b Fg(adimensional)e(momentum)f(distanc)n(e)f (fr)n(om)g(the)g(F)l(ermi)h(sur-)109 2071 y(fac)n(e)46 2189 y Fp(6.)25 b Fm(j)p Fn(x)p Fm(j)13 b Fp(=)h Fn(a)256 2171 y Fi(\000)p Fj(1)303 2138 y Fk(q)p 344 2138 201 2 v 344 2189 a Fn(x)372 2172 y Fj(2)372 2200 y(0)391 2189 y Fn(v)417 2172 y Fj(2)415 2200 y(0)448 2189 y Fp(+)c Fn(~)-23 b(x)525 2175 y Fj(2)544 2189 y Fp(:)21 b Fg(adimensional)e (length)-13 2316 y Fd(Theorem)d(1)j(\(main)e(result\))23 b Fg(Consider)18 b(a)f(F)l(ermi)g(system)h(with)g(hamiltonian)490 2402 y Fl(n)470 2415 y Fk(X)494 2506 y Fl(i)530 2456 y Fp(\()p Fm(\000)593 2421 y Fp(\026)-25 b Fn(h)621 2399 y Fj(2)640 2421 y Fn(@)669 2402 y Fj(2)666 2433 y Fl(~)-18 b(x)686 2438 y Fe(i)p 593 2445 109 2 v 614 2490 a Fp(2)p Fn(m)717 2456 y Fp(+)11 b Fn(uc)p Fp(\()o Fn(~)-23 b(x)862 2463 y Fl(i)876 2456 y Fp(\))11 b Fm(\000)g Fn(\026)p Fp(\))g(+)g(2)p Fn(\025)1124 2415 y Fk(X)1127 2506 y Fl(i)h Fp(\(\001\()p Fn(u)p Fp(\)\))860 330 y Fi(\000)p Fj(1)568 488 y Fm(j)p Fn(S)612 495 y Fj(0)632 488 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fn(;)g Fm(j)789 475 y Fp(~)782 488 y Fn(S)r Fp(\()p Fn(x;)g(y)r Fp(\))p Fm(j)k(\024)1088 454 y Fp(\001\()p Fn(u)p Fp(\))p Fn(C)1230 461 y Fl(N)1263 454 y Fn(a)1289 436 y Fi(\000)p Fj(1)p 1012 476 400 2 v 1012 522 a Fp(1)f(+)g(\001\()p Fn(u)p Fp(\))1203 508 y Fl(N)1236 522 y Fm(j)p Fn(x)g Fm(\000)g Fn(y)r Fm(j)1379 508 y Fl(N)1790 488 y Fp(\(16\))109 626 y Fg(wher)n(e)k Fn(C)279 633 y Fl(N)327 626 y Fg(is)g(a)g(suitable) h(c)n(onstant.)23 b(The)15 b(o)n(c)n(cup)n(ation)g(numb)n(er)g(disc)n (ontinuity)g(e)n(q.\(6\))g(is)g(such)109 704 y(that)i Fn(Z)245 686 y Fi(\000)p Fj(1)306 704 y Fp(=)d(1)e(+)f Fn(O)q Fp(\()510 685 y Fl(\025)p 505 692 30 2 v 505 721 a Fj(\001)540 704 y Fp(\))p Fg(.)-43 824 y Fp(case)16 b(2)p -43 831 128 2 v 25 w Fg(\()p Fn(\025)24 b Fg(and)g(the)g(gap)g (smal)r(l\),)i(for)d Fn(u)i(<)g(")e Fg(ther)n(e)h(ar)n(e)f(two)h(r)n(e) n(gular)f(functions)28 b Fp(^)-27 b Fn(u)p Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))23 b Fg(and)119 890 y Fp(^)109 902 y Fn(Z)146 884 y Fi(\000)p Fj(1)193 902 y Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))14 b Fg(such)h(that)f Fm(j)s Fp(^)-27 b Fn(u)p Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))t Fm(\000)t Fn(u)p Fm(j)k Fp(=)i Fn(O)q Fp(\()p Fn(\025)p Fp(\))p Fg(,)i Fm(j)1095 890 y Fp(^)1085 902 y Fn(Z)1122 884 y Fi(\000)p Fj(1)1169 902 y Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))t Fm(\000)t Fp(1)p Fm(j)13 b Fp(=)h Fn(O)q Fp(\()p Fn(\025)p Fp(\))h Fg(for)f Fm(j)p Fn(k)r Fm(j)g Fn(>)f(\031)r(=)p Fp(2)109 980 y Fg(and)399 1059 y Fp(lim)387 1090 y Fi(j)p Fl(k)q Fi(j!)p Fj(0)490 1059 y Fp(^)-27 b Fn(u)o Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))14 b(=)g Fn(u)773 1038 y Fj(1)p Fi(\000)p Fl(\021)836 1043 y Ff(2)853 1038 y Fj(\()p Fl(\025;u)p Fj(\))1053 1059 y Fp(lim)1041 1090 y Fi(j)p Fl(k)q Fi(j!)p Fj(0)1151 1046 y Fp(^)1141 1059 y Fn(Z)1178 1038 y Fi(\000)p Fj(1)1225 1059 y Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))14 b(=)g Fn(u)1484 1038 y Fl(\021)1502 1043 y Ff(1)1518 1038 y Fj(\()p Fl(\025;u)p Fj(\))109 1174 y Fg(wher)n(e)k Fn(\025)275 1155 y Fi(\000)p Fj(2)323 1174 y Fn(\021)347 1181 y Fj(1)367 1174 y Fp(\()p Fn(\025;)8 b(u)p Fp(\))18 b Fg(and)g Fn(\025)624 1155 y Fi(\000)p Fj(1)672 1174 y Fn(\021)696 1181 y Fj(2)716 1174 y Fp(\()p Fn(\025;)8 b(u)p Fp(\))18 b Fg(b)n(oth)g(p)n(ositive)g (and)h(indep)n(endent)h(of)e Fn(u)g Fg(in)h(the)f Fn(\025)e Fm(!)f Fp(0)109 1252 y Fg(limit,)h(such)g(that,)g(for)f(a)h(suitable)h Fn(\027)g Fp(=)d Fn(\027)s Fp(\()p Fn(\025;)8 b(u)p Fp(\))14 b(=)f Fn(O)q Fp(\()p Fn(\025)1138 1234 y Fj(2)1159 1252 y Fp(\))p Fg(,)j(the)g(function)h Fn(S)s Fp(\()p Fn(x;)8 b(y)r Fp(\))15 b Fg(is)g(given)j(by,)670 1380 y Fn(S)s Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)h Fn(S)912 1387 y Fl(A)940 1380 y Fp(\()p Fn(x;)8 b(y)r Fp(\))i(+)h Fn(\025S)1171 1387 y Fl(B)1202 1380 y Fp(\()p Fn(x;)d(y)r Fp(\))474 b(\(17\))109 1508 y Fg(with)302 1636 y Fn(S)332 1643 y Fl(A)360 1636 y Fp(\()p Fn(x;)8 b(y)r Fp(\))41 b(=)594 1577 y Fk(Z)644 1636 y Fn(dk)r(\036)p Fp(\()o Fn(~)-23 b(x;)8 b(k)r(;)j Fp(^)-27 b Fn(u)o Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\)\))p Fn(\036)p Fp(\()p Fm(\000)o Fn(~)-23 b(y)q(;)8 b(k)r(;)j Fp(^)-27 b Fn(u)p Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\)\))p Fn(e)1472 1615 y Fl(ik)1502 1620 y Ff(0)1519 1615 y Fj(\()p Fl(x)1553 1620 y Ff(0)1570 1615 y Fi(\000)p Fl(y)1614 1620 y Ff(0)1632 1615 y Fj(\))1659 1636 y Fm(\001)117 b Fp(\(18\))688 1718 y(1)p 599 1740 202 2 v 609 1780 a(^)599 1793 y Fn(Z)t Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))1158 1718 y(1)p 827 1740 686 2 v 827 1788 a Fm(\000)p Fn(ik)908 1795 y Fj(0)939 1788 y Fp(+)j(\()p Fn(")p Fp(\()p Fn(k)r(;)g Fp(^)-27 b Fn(u)o Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\)\))j Fm(\000)g Fn(\026)1399 1795 y Fj(0)1419 1788 y Fp(\)\026)-25 b Fn(h)1466 1767 y Fi(\000)p Fj(1)1790 1859 y Fp(\(19\))109 1987 y Fg(The)18 b(o)n(c)n(cup)n(ation)f(numb)n(er)h(disc)n(ontinuity)g (e)n(q.\(6\))g(is)g Fn(Z)1141 1969 y Fi(\000)p Fj(1)1202 1987 y Fp(=)d Fn(u)1283 1969 y Fl(\021)1301 1974 y Ff(1)1320 1987 y Fp(\(1)c(+)h Fn(O)q Fp(\()p Fn(\025)p Fp(\)\))18 b Fg(and)g(for)g(any)f Fn(N)109 2065 y Fg(and)g(for)g Fm(j)p Fn(x)11 b Fm(\000)g Fn(y)r Fm(j)i Fn(>)h(u)517 2047 y Fi(\000)p Fj(\(1)p Fi(\000)p Fl(\021)621 2052 y Ff(2)638 2047 y Fj(\))671 2065 y Fg(we)k(have)529 2208 y Fm(j)p Fn(S)573 2215 y Fl(A)601 2208 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fm(j)p Fn(;)g Fm(j)p Fn(S)795 2215 y Fl(B)824 2208 y Fp(\()p Fn(x;)g(y)r Fp(\))p Fm(j)13 b(\024)1030 2175 y Fn(Z)1067 2157 y Fi(\000)p Fj(1)1114 2175 y Fn(u)1142 2157 y Fj(1)p Fi(\000)p Fl(\021)1205 2162 y Ff(2)1222 2157 y Fj(\()p Fl(\025;u)p Fj(\))1302 2175 y Fn(C)1337 2182 y Fl(N)1371 2175 y Fn(a)1397 2157 y Fi(\000)p Fj(1)p 1023 2197 429 2 v 1023 2243 a Fp(1)e(+)g Fn(u)1135 2228 y Fj(\(1)p Fi(\000)p Fl(\021)1212 2233 y Ff(2)1229 2228 y Fj(\))p Fl(N)1276 2243 y Fm(j)p Fn(x)g Fm(\000)f Fn(y)r Fm(j)1418 2228 y Fl(N)1790 2208 y Fp(\(20\))109 2344 y Fg(wher)n(e)17 b Fn(C)281 2351 y Fl(N)332 2344 y Fg(is)h(a)f (suitable)i(c)n(onstant.)10 2476 y Fp(F)l(or)e(generalit)o(y)e(the)i (second)f(statemen)o(t)f(of)i(the)f(theorem)f(is)i(written)f(for)g(an)o (y)h(v)m(alue)f(of)h(the)f(ratio)-8 2534 y Fl(\025)p -8 2542 21 2 v -8 2571 a(u)17 2554 y Fp(;)j(ho)o(w)o(ev)o(er)f(for)g Fm(j)p Fn(\025)p Fm(j)g Fn(<)g("u)g Fp(the)g(t)o(w)o(o)g(statemen)o(ts) f(o)o(v)o(erlap)h(and)h(of)f(course)h(they)f(agree,)h(as)g(in)f(this) 919 2728 y(9)p eop %%Page: 10 10 10 9 bop -13 -5 a Fp(case)21 b(^)-27 b Fn(u)p Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))17 b(=)g Fn(u)p Fp(\(1)c(+)496 -24 y Fl(\025)p 496 -16 21 2 v 496 12 a(u)522 -5 y Fn(c)543 2 y Fj(1)563 -5 y Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\)\),)789 -17 y(^)779 -5 y Fn(Z)t Fp(\()p Fn(k)r(;)g(\025;)g(u)p Fp(\))17 b(=)g Fn(u)p Fp(\(1)c(+)1192 -24 y Fl(\025)p 1192 -16 V 1192 12 a(u)1218 -5 y Fn(c)1239 2 y Fj(2)1259 -5 y Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\)\))18 b(with)g Fn(c)1595 2 y Fj(1)1615 -5 y Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))18 b(and)-13 74 y Fn(c)8 81 y Fj(2)28 74 y Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))16 b(b)q(ounded.)10 152 y(The)e(pro)q(of)h(of)f(the)f(theorem)f(is)h(p)q(erformed)g(b)o(y)g (using)h(renormalization)e(group)i(tec)o(hniques,)e(whic)o(h)-13 230 y(pro)o(vide)19 b(us)h(with)f(an)h(algorithm)f(to)h(express)f(the)h 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Fp(\()o Fn(~)-23 b(x;)8 b(k)r(;)g(u)p Fp(\))p Fn(\036)p Fp(\()p Fm(\000)o Fn(~)-23 b(y;)8 b(k)r(;)g(u)p Fp(\))p Fn(e)975 841 y Fl(ik)1005 846 y Ff(0)1022 841 y Fj(\()p Fl(x)1056 846 y Ff(0)1073 841 y Fi(\000)p Fl(y)1117 846 y Ff(0)1135 841 y Fj(\))1156 828 y Fn(f)1185 810 y Fl(h)1208 828 y Fp(\()p Fn(k)1254 810 y Fj(2)1252 840 y(0)1285 828 y Fp(+)j(\()p Fn(")p Fp(\()1392 815 y Fn(~)1395 828 y(k)q(;)d(u)p Fp(\))j Fm(\000)g Fn(\026)1580 835 y Fj(0)1600 828 y Fp(\))1619 810 y Fj(2)1639 828 y Fp(\))p 1156 850 502 2 v 1203 904 a Fm(\000)p Fn(ik)1284 911 y Fj(0)1314 904 y Fp(+)g Fn(")p Fp(\()1402 891 y Fn(~)1405 904 y(k)r(;)d(u)p Fp(\))j Fm(\000)g Fn(\026)1591 911 y Fj(0)-13 1006 y Fp(with)g Fn(f)122 988 y Fj(1)142 1006 y Fp(\()p Fn(t)p Fp(\))f(is)h(a)g Fn(C)326 988 y Fi(1)374 1006 y Fp(function)g(whic)o(h) f(is)h(equal)g(to)g(1)g(for)h Fm(j)p Fn(t)p Fm(j)h(\025)g Fn(\015)s Fp(,)f Fn(f)1215 988 y Fl(h)1238 1006 y Fp(\()p Fn(t)p Fp(\),)f(for)g Fn(h)j Fm(6)p Fp(=)g(1,)e(is)f(a)g Fn(C)1650 988 y Fi(1)1698 1006 y Fp(compact)-13 1084 y(supp)q(ort)19 b(function)f(with)g(supp)q(ort)h(in)e Fn(\015)743 1066 y Fl(h)783 1084 y Fm(\024)f(j)p Fn(t)p Fm(j)g(\024)h Fn(\015)984 1066 y Fl(h)p Fj(+2)1069 1084 y Fp(and)1166 1051 y Fk(P)1210 1094 y Fj(1)p Fi(\024)p Fl(h<)p Fi(\0001)1376 1084 y Fn(f)1405 1066 y Fl(h)1428 1084 y Fp(\()p Fn(t)p Fp(\))f Fm(\021)g Fp(1)j(\(see)e(the)h(next)-13 1162 y(section\).)i(Of)13 b(course)h Fn(g)425 1144 y Fl(h)448 1162 y Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b Fm(\021)g Fp(0)i(for)f Fn(h)g Fm(\024)f Fn(h)860 1144 y Fi(\003)894 1162 y Fp(with)h Fn(\015)1031 1144 y Fl(h)1051 1132 y Fh(\003)1085 1162 y Fm(')f Fp(\001\()p Fn(u)p Fp(\))h(as)g(min)1395 1169 y Fl(~)1396 1178 y(k)1426 1162 y Fm(j)p Fn(")p Fp(\()1479 1149 y Fn(~)1482 1162 y(k)r(;)8 b(u)p Fp(\))e Fm(\000)g Fn(\026)1658 1169 y Fj(0)1677 1162 y Fm(j)14 b Fp(=)g(\001\()p Fn(u)p Fp(\),)-13 1240 y Fg(i.e.)j Fp(the)d(Sc)o(h)o(winger)e(function) h(is)h(expressed)e(in)h(terms)f(of)i(a)g(\014nite)f(sum)f(of)i (functions.)20 b(In)13 b(App)q(endix)-13 1319 y(1,2)j(it)g(is)g(sho)o (wn)h(that,)f(for)h(an)o(y)f(N:)523 1452 y Fm(j)p Fn(g)562 1431 y Fl(h)585 1452 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fm(j)p Fn(;)g Fm(j)r Fp(^)-26 b Fn(g)774 1431 y Fl(h)795 1452 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fm(j)13 b(\024)1068 1418 y Fn(\015)1096 1400 y Fl(h)1119 1418 y Fn(C)1154 1425 y Fl(N)1187 1418 y Fn(a)1213 1400 y Fi(\000)p Fj(1)p 993 1440 342 2 v 993 1486 a Fp(1)f(+)f Fn(\015)1106 1471 y Fl(hN)1160 1486 y Fm(j)p Fn(x)f Fm(\000)h Fn(y)r Fm(j)1302 1471 y Fl(N)-13 1581 y Fp(so)17 b(that)f(the)g(\014rst)h(statemen)o(t)d (of)j(the)f(theorem)f(follo)o(ws.)10 1660 y(In)d(the)f(second)g(case,)h (in)f(whic)o(h)g(the)g(in)o(teraction)g(and)h(the)f(gap)h(amplitude)e (are)h(small)f(and)i Fn(u)i(<<)f Fm(j)p Fn(\025)p Fm(j)-13 1738 y Fp(for)j(simplicit)o(y)-5 b(,)13 b(w)o(e)j(ha)o(v)o(e)530 1860 y Fn(S)s Fp(\()p Fn(x;)8 b(y)r Fp(\))k(=)789 1819 y Fk(X)741 1911 y Fj(1)p Fi(\025)p Fl(h>)p Fi(\0001)897 1860 y Fp([)p Fn(g)936 1840 y Fl(h)958 1860 y Fp(\()p Fn(x;)c(y)r Fp(\))i(+)h Fn(\025)r Fp(\026)-26 b Fn(g)1184 1840 y Fl(h)1207 1860 y Fp(\()p Fn(x;)8 b(y)r Fp(\)])455 b(\(22\))-13 2004 y(with)131 2135 y Fn(g)156 2114 y Fl(h)178 2135 y Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)378 2101 y(1)p 362 2123 56 2 v 362 2169 a Fn(Z)395 2176 y Fl(h)431 2076 y Fk(Z)481 2135 y Fn(dk)r(\036)p Fp(\()o Fn(~)-23 b(x;)8 b(k)r(;)g(\033)708 2142 y Fl(h)730 2135 y Fp(\))p Fn(\036)p Fp(\()p Fm(\000)o Fn(~)-23 b(y)q(;)8 b(k)r(;)g(\033)960 2142 y Fl(h)982 2135 y Fp(\))p Fn(e)1024 2114 y Fl(ik)1054 2119 y Ff(0)1071 2114 y Fj(\()p Fl(x)1105 2119 y Ff(0)1122 2114 y Fi(\000)p Fl(y)1166 2119 y Ff(0)1184 2114 y Fj(\))1205 2101 y Fn(f)1234 2083 y Fl(k)1255 2101 y Fp(\()p Fn(k)1301 2083 y Fj(2)1299 2113 y(0)1332 2101 y Fp(+)j(\()p Fn(")p Fp(\()1439 2088 y Fn(~)1442 2101 y(k)r(;)d(\033)1519 2108 y Fl(h)1541 2101 y Fp(\))j Fm(\000)g Fn(\026)1650 2108 y Fj(0)1670 2101 y Fp(\))1689 2083 y Fj(2)1709 2101 y Fp(\))p 1205 2123 523 2 v 1251 2177 a Fm(\000)p Fn(ik)1332 2184 y Fj(0)1363 2177 y Fp(+)g Fn(")p Fp(\()1451 2164 y Fn(~)1454 2177 y(k)q(;)d(\033)1530 2184 y Fl(h)1552 2177 y Fp(\))j Fm(\000)g Fn(\026)1661 2184 y Fj(0)-13 2281 y Fp(with)22 b Fn(Z)137 2288 y Fl(h)183 2281 y Fm(')h Fn(\015)273 2263 y Fi(\000)p Fl(\021)318 2268 y Ff(1)336 2263 y Fj(\()p Fl(\025;u)p Fj(\))p Fl(h)459 2281 y Fp(and)f Fn(\033)587 2288 y Fl(h)633 2281 y Fm(')h Fn(u\015)751 2263 y Fi(\000)p Fl(\021)796 2268 y Ff(2)813 2263 y Fj(\()p Fl(\025;u)p Fj(\))p Fl(h)914 2281 y Fp(.)39 b(Again)22 b(the)f(Sc)o(h)o(winger)g(function)h(is)g(giv)o(en)f(b)o(y)-13 2359 y(a)f(\014nite)g(sum)f(but)h(no)o(w)h(the)f(\\last)g(scale")g Fn(h)847 2341 y Fi(\003)887 2359 y Fp(is)g(giv)o(en)f(b)o(y)h(the)g (condition)g Fn(\015)1477 2341 y Fl(h)1497 2329 y Fh(\003)1538 2359 y Fm(')g Fn(\033)1625 2366 y Fl(h)1645 2357 y Fh(\003)1685 2359 y Fg(i.e.)e Fn(h)1798 2341 y Fi(\003)1838 2359 y Fm(')-13 2437 y Fp(\(1)11 b Fm(\000)g Fn(\021)115 2444 y Fj(2)135 2437 y Fp(\()p Fn(\025;)d(u)p Fp(\)\))g(log)q(\()p Fn(u)p Fp(\).)21 b(Since)15 b(for)i(all)f Fn(N)500 2573 y Fm(j)p Fn(g)539 2552 y Fl(h)561 2573 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fm(j)p Fn(;)g Fm(j)r Fp(\026)-26 b Fn(g)750 2552 y Fl(h)771 2573 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fm(j)13 b(\024)985 2539 y Fp(1)p 970 2561 56 2 v 970 2607 a Fn(Z)1003 2614 y Fl(h)1101 2539 y Fn(\015)1129 2521 y Fl(h)1152 2539 y Fn(C)1187 2546 y Fl(N)1220 2539 y Fn(a)1246 2521 y Fi(\000)p Fj(1)p 1035 2561 324 2 v 1035 2607 a Fp(1)f(+)f Fn(\033)1150 2590 y Fl(N)1148 2620 y(h)1183 2607 y Fm(j)p Fn(x)g Fm(\000)f Fn(y)r Fm(j)1325 2593 y Fl(N)1790 2573 y Fp(\(23\))907 2728 y(10)p eop %%Page: 11 11 11 10 bop -13 -5 a Fp(the)18 b(second)g(statemen)o(t)f(of)h(the)g (theorem)f(follo)o(ws.)27 b(Note)17 b(that)22 b(^)-27 b Fn(u)p Fp(\()p Fn(k)r(;)8 b(\025;)g(u)p Fp(\))18 b(in)g(the)g (theorem)e(is)i(not)-13 74 y(completely)c(determined)h(as)j(for)g(a)g (small)e Fn(\025)i(\036)p Fp(\()888 60 y Fn(~)891 74 y(k)r(;)7 b(~)-23 b(x)o(;)8 b(u)p Fp(\(1)k(+)g Fn(\025)p Fp(\)\))k(=)g Fn(\036)p Fp(\()1303 60 y Fn(~)1306 74 y(k)r(;)7 b(~)-23 b(x;)8 b(u)p Fp(\))j(+)h Fn(\025)1547 60 y Fp(~)1541 74 y Fn(\036)p Fp(\()1586 60 y Fn(~)1589 74 y(k)r(;)7 b(~)-23 b(x;)8 b(u)p Fp(\))17 b(where)-7 139 y(~)-13 152 y Fn(\036)p Fp(\()32 139 y Fn(~)35 152 y(k)r(;)7 b(~)-23 b(x)o(;)8 b(u)p Fp(\),)13 b Fn(\036)p Fp(\()252 139 y Fn(~)255 152 y(k)r(;)7 b(~)-23 b(x)o(;)8 b(u)p Fp(\))13 b(and)g(all)f(their)g(deriv)m(ativ)o(es)g(ob)q(eys)h(to) g(the)g(same)e(b)q(ounds)k(and)e(the)f(same)g(holds)-13 230 y(for)h Fn(")p Fp(\()p Fn(k)r(;)8 b(u)p Fp(\))k(\(see)h(App)q (endix)f(2\))h(so)h(that)f(w)o(e)g(can)g(replace)i(^)-27 b Fn(u)13 b Fp(in)f(eq.\(18\))h(with)j(^)-27 b Fn(u)o Fp(\(1)t(+)t Fn(O)q Fp(\()p Fn(\025)p Fp(\)\))15 b(b)o(y)e(simply)-13 308 y(replacing)i Fn(S)225 315 y Fl(B)256 308 y Fp(\()p Fn(x;)8 b(y)r Fp(\))15 b(with)h(a)h(prop)q(er)701 296 y(~)693 308 y Fn(S)723 315 y Fl(B)753 308 y Fp(\()p Fn(x;)8 b(y)r Fp(\))15 b(ob)q(eing)i(to)g(the)f(same)f(b)q(ound.)10 387 y(If)j Fn(u)f Fm(!)f Fp(0)j(than)f Fn(h)358 368 y Fi(\003)395 387 y Fm(!)f(1)g Fp(and,)i(from)e(eq.\(22\),)h(w)o(e)f (\014nd)i(that)f Fn(S)s Fp(\()p Fn(x;)8 b(y)r Fp(\))17 b(asymptotically)f(b)q(eha)o(v)o(es)-13 465 y(as)h Fm(j)p Fn(x)10 b Fm(\000)h Fn(y)r Fm(j)i(!)h(1)i Fp(as)425 442 y Fl(S)446 447 y Ff(0)464 442 y Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))p 397 453 188 2 v 397 485 a Fi(j)p Fl(x)p Fi(\000)p Fl(y)q Fi(j)482 474 y Fe(\021)498 481 y Ff(1)516 474 y(\()p Fe(\025;)p Ff(0\))590 465 y Fp(,)g(so)h(tha)f(the)g(results) g(of)h(\(B.G.P)l(.S.\))d(are)j(reco)o(v)o(ered.)10 543 y(A)o(t)23 b(the)h(end)g(of)g(this)g(pap)q(er)g(w)o(e)g(giv)o(e)e(some) h(argumen)o(ts)g(supp)q(orting)i(the)f(conjecture)f(that)h(a)-13 621 y(similar)15 b(b)q(eha)o(viour)j(is)f(found)h(also)g(in)f(the)g (spinning)h(case,)f Fg(if)h(the)h(inter)n(action)h(is)e(either)h(r)n (epulsive)-13 700 y(or)j(attr)n(active)h(but)h Fn(u)f Fm(\025)g Fn(k)502 707 y Fj(1)522 700 y Fn(e)545 681 y Fi(\000j)p Fl(\025)p Fi(j)613 670 y Fh(\000)p Fe(k)653 677 y Ff(2)697 700 y Fg(for)f(some)h(p)n(ositive)f(c)n(onstants)i Fn(k)1335 707 y Fj(1)1355 700 y Fn(;)8 b(k)1402 707 y Fj(2)1422 700 y Fp(.)38 b(This)22 b(suggests)h(that)-13 778 y(the)16 b Fn(u)d Fp(=)h(0)i(spinning)h(case)f(with)g(attractiv)o (e)f(in)o(teraction)g(generates)h(sp)q(on)o(taneously)h(at)f(the)g(F)l (ermi)-13 856 y(surface)g(a)h(gap)g Fn(O)q Fp(\()p Fn(e)364 838 y Fi(\000j)p Fl(\025)p Fi(j)432 826 y Fh(\000)p Fe(k)472 833 y Ff(2)494 856 y Fp(\))f(ev)o(en)f(in)h(absence)g(of)g(a)h(p)q (erio)q(dic)f(p)q(oten)o(tial.)10 934 y(F)l(or)h(simplicit)n(y)c(in)j (the)g(follo)o(wing)g(w)o(e)g(use)g(dimensionless)e(quan)o(tities)i (de\014ned)g(as:)246 1082 y(~)-27 b Fn(x)14 b Fp(=)g Fn(x=a)513 1073 y Fp(~)512 1082 y Fn(t)f Fp(=)h(\026)-25 b Fn(h)q(t=)p Fp(2)p Fn(ma)758 1061 y Fj(2)887 1069 y Fp(~)875 1082 y Fn(H)18 b Fp(=)c(2)p Fn(ma)1078 1061 y Fj(2)1097 1082 y Fn(H)q(=)p Fp(\026)-25 b Fn(h)1190 1061 y Fj(2)1318 1069 y Fp(~)1308 1082 y Fn( )1342 1061 y Fl(")1340 1094 y(~)-18 b(x)o(;\033)1406 1082 y Fp(=)1458 1001 y Fk(s)p 1500 1001 36 2 v 1506 1048 a Fp(1)p 1505 1070 26 2 v 1505 1116 a Fn(a)1535 1082 y( )1569 1061 y Fl(")1567 1094 y(~)g(x;\033)1790 1082 y Fp(\(24\))-13 1224 y(so)20 b(that)g(the)g(hamiltonian)534 1211 y(~)521 1224 y Fn(H)k Fp(is)c(giv)o(en)f(b)o(y)g(eq.\(1\))g(with)1098 1211 y(\026)1097 1224 y Fn(h)h Fp(=)g Fn(a)f Fp(=)h(1)p Fn(;)8 b Fp(2)p Fn(m)20 b Fp(=)g(1)p Fn(;)8 b(\026)20 b Fp(=)g Fn(\031)1680 1206 y Fj(2)1699 1224 y Fp(.)31 b(In)20 b(the)-13 1302 y(follo)o(wing)h(the)g(tilde's)f(will)g(b)q(e)h (omitted)f(and)i(w)o(e)e(consider)h(directly)f(dimensionless)f(quan)o (tities.)-13 1380 y(Moreo)o(v)o(er)c(w)o(e)h(write)f(simply)f Fn(")p Fp(\()592 1367 y Fn(~)595 1380 y(k)r Fp(\))i(for)h Fn(")p Fp(\()771 1367 y Fn(~)774 1380 y(k)r(;)8 b(u)p Fp(\))16 b(and)g(\001)g(for)h(\001\()p Fn(u)p Fp(\))e(unless)i (explicitely)c(stated.)-13 1574 y Fo(2)83 b(Pro)r(of)28 b(of)g(the)g(theorem)-13 1720 y Fc(2.1)69 b(Multiscale)20 b(decomp)r(ositi)o(on)g(and)k(the)e(lo)r(calization)e(op)r(erator)-13 1831 y Fp(In)11 b(this)h(section)f(w)o(e)h(pro)o(v)o(e)f(the)g(second)i (statemen)o(t)d(of)i(the)f(theorem,)g(whic)o(h)g(is)h(the)f(more)g(tec) o(hnically)-13 1909 y(in)o(v)o(olv)o(ed)21 b(one;)27 b(the)c(pro)q(of)h(of)g(the)f(\014rst)h(statemen)o(t)e(is)h(easier)g (and)h(it)f(is)g(in)g(App)q(endix)g(1.)43 b(W)l(e)-13 1987 y(consider)17 b(a)h(Grassman)g(algebra,)g(whose)g(elemen)o(ts)c Fn( )1015 1969 y Fl(")1013 2000 y(k)q(;\033)1066 1987 y Fp(,)j Fn(")f Fp(=)g Fm(\006)p Fp(,)h(v)o(erify)f Fm(f)p Fn( )1456 1969 y Fl(")1454 2000 y(k)q(;\033)1506 1987 y Fn(;)8 b( )1562 1969 y Fl(")1578 1958 y Fh(0)1560 2000 y Fl(k)1579 1990 y Fh(0)1590 2000 y Fl(;\033)1621 1990 y Fh(0)1634 1987 y Fm(g)16 b Fp(=)g(0.)26 b(The)-13 2066 y Fg(Euclide)n(an)d(\014elds)f Fp(can)f(b)q(e)g(de\014ned)f(as)i Fn( )779 2048 y Fl(")777 2078 y(x;\033)851 2066 y Fp(=)911 2030 y Fk(R)947 2066 y Fn(dk)r(e)1022 2048 y Fl(i")p Fj(\()p Fl(k)1082 2053 y Ff(0)1100 2048 y Fl(x)1120 2053 y Ff(0)1137 2048 y Fj(+)1162 2039 y Fl(~)1164 2048 y(k)q(~)-17 b(x)o Fj(\))1219 2066 y Fn( )1253 2048 y Fl(")1251 2078 y(k)q(;\033)1324 2066 y Fp(and)1424 2030 y Fk(R)1460 2066 y Fn(dk)24 b Fp(=)1599 2042 y Fj(\(2)p Fl(\031)q Fj(\))1666 2031 y Ff(2)p 1599 2054 84 2 v 1614 2083 a Fl(\014)r(N)1696 2032 y Fk(P)1740 2076 y Fl(k)1782 2066 y Fp(with)-13 2144 y Fn(e)10 2126 y Fl(ik)40 2131 y Ff(0)57 2126 y Fl(\014)95 2144 y Fp(=)13 b Fm(\000)p Fp(1)p Fn(;)8 b(e)254 2126 y Fl(i)264 2117 y(~)266 2126 y(k)q(N)333 2144 y Fp(=)13 b(1)k(and)g(the)f(grassmanian)g(in)o(tegration)g(is)g (de\014ned)g(on)h(monomials)d(b)o(y:)329 2213 y Fk(Z)379 2272 y Fn(P)7 b Fp(\()p Fn(d )r Fp(\))p Fn( )548 2251 y Fj(+)546 2284 y Fl(x)566 2289 y Ff(1)593 2272 y Fm(\001)h(\001)g (\001)h Fn( )694 2251 y Fj(+)692 2284 y Fl(x)712 2288 y Fe(n)735 2272 y Fn( )769 2251 y Fi(\000)767 2284 y Fl(y)784 2289 y Ff(1)811 2272 y Fm(\001)f(\001)g(\001)h Fn( )912 2251 y Fi(\000)910 2284 y Fl(y)927 2288 y Fe(n)964 2272 y Fp(=)1016 2230 y Fk(X)1035 2318 y Fl(\031)1076 2272 y Fp(\()p Fm(\000)p Fp(1\))1177 2251 y Fl(\033)1197 2255 y Fe(\031)1229 2230 y Fk(Y)1249 2322 y Fl(i)1290 2272 y Fn(g)r Fp(\()p Fn(x)1362 2279 y Fl(i)1387 2272 y Fm(\000)i Fn(y)1461 2280 y Fl(\031)q Fj(\()p Fl(i)p Fj(\))1523 2272 y Fp(\))248 b(\(25\))-13 2417 y(where)16 b Fn(\031)h Fp(is)f(a)h(p)q(erm)o(utation)e(of)i(the)f(set)g(\(1)p Fn(;)8 b(:)g(:)g(:)g(;)g(n)p Fp(\),)15 b Fn(\033)1016 2424 y Fl(\031)1056 2417 y Fp(is)h(the)g(parit)o(y)g(of)g(the)g(p)q (erm)o(utation)f(and)595 2560 y Fn(g)r Fp(\()p Fn(x)c Fm(\000)g Fn(y)r Fp(\))j(=)838 2502 y Fk(Z)888 2560 y Fn(dk)1016 2526 y(e)1039 2508 y Fi(\000)p Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))p 946 2549 318 2 v 946 2602 a Fm(\000)p Fn(ik)1027 2609 y Fj(0)1057 2602 y Fp(+)1104 2589 y Fn(~)1106 2602 y(k)1133 2588 y Fj(2)1164 2602 y Fm(\000)d Fn(\031)1244 2588 y Fj(2)907 2728 y Fp(11)p eop %%Page: 12 12 12 11 bop -13 -5 a Fp(An)22 b(application)f(of)i(T)l(rotter's)f(form)o (ula)e(allo)o(ws)i(us,)i(as)e(usual)h(\(see)e(\(B.G.\)\),)h(to)g(write) g(the)g(t)o(w)o(o)-13 74 y(p)q(oin)o(ts)16 b(Sc)o(h)o(winger)g (function)g(in)g(terms)e(of)j(grassmanian)g(in)o(tegral)e Fn(P)7 b Fp(\()p Fn(d )r Fp(\))16 b(as:)553 223 y Fn(S)s Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)29 b(lim)774 247 y Fe(\014)q Fh(!1)770 275 y Fe(N)s Fh(!1)876 150 y Fk(R)912 185 y Fn(P)7 b Fp(\()p Fn(d )r Fp(\))p Fn(e)1070 167 y Fi(\000)1102 159 y Fj(\026)1097 167 y Fl(V)g Fj(\()p Fl( )q Fj(\))1179 185 y Fn( )1213 167 y Fj(+)1211 198 y Fl(x)1242 185 y Fn( )1276 167 y Fi(\000)1274 198 y Fl(y)p 876 211 430 2 v 939 222 a Fk(R)975 258 y Fn(P)g Fp(\()p Fn(d )r Fp(\))p Fn(e)1133 243 y Fi(\000)1165 235 y Fj(\026)1160 243 y Fl(V)g Fj(\()p Fl( )q Fj(\))1790 223 y Fp(\(26\))-13 374 y(where)135 361 y(\026)128 374 y Fn(V)k Fp(\()p Fn( )r Fp(\))16 b(is)g(giv)o(en)f(b)o(y)506 361 y(\026)499 374 y Fn(V)c Fp(\()p Fn( )r Fp(\))i(=)h Fn(\025V)23 b Fp(+)11 b Fn(\027)s(N)17 b Fp(+)11 b Fn(uP)23 b Fp(with:)297 502 y Fn(V)i Fp(=)403 460 y Fk(X)402 553 y Fl(\033)o(;\033)452 543 y Fh(0)472 443 y Fk(Z)495 538 y Fj(\003)p Fi(\002)p Fj(\003)582 502 y Fn(dx)635 509 y Fj(1)655 502 y Fn(dx)708 509 y Fj(2)727 502 y Fn(v)r Fp(\()o Fn(~)-23 b(x)800 509 y Fj(1)830 502 y Fm(\000)10 b Fn(~)-23 b(x)908 509 y Fj(2)927 502 y Fp(\))p Fn(\016)r Fp(\()p Fn(x)1017 509 y Fj(0)p Fl(;)p Fj(1)1075 502 y Fm(\000)11 b Fn(x)1153 509 y Fj(0)p Fl(;)p Fj(2)1200 502 y Fp(\))p Fn( )1253 481 y Fj(+)1251 514 y Fl(x)1271 519 y Ff(1)1287 514 y Fl(;\033)1321 502 y Fn( )1355 481 y Fj(+)1353 515 y Fl(x)1373 520 y Ff(2)1389 515 y Fl(;\033)1420 505 y Fh(0)1434 502 y Fn( )1468 481 y Fi(\000)1466 515 y Fl(x)1486 520 y Ff(2)1503 515 y Fl(;\033)1534 505 y Fh(0)1547 502 y Fn( )1581 481 y Fi(\000)1579 514 y Fl(x)1599 519 y Ff(1)1616 514 y Fl(;\033)297 639 y Fn(N)19 b Fp(=)407 597 y Fk(X)426 684 y Fl(\033)475 580 y Fk(Z)498 674 y Fj(\003)533 639 y Fn(dx )620 618 y Fj(+)618 651 y Fl(x;\033)671 639 y Fn( )705 618 y Fi(\000)703 651 y Fl(x;\033)853 639 y Fn(P)i Fp(=)957 597 y Fk(X)976 684 y Fl(\033)1025 580 y Fk(Z)1049 674 y Fj(\003)1083 639 y Fn(dxc)p Fp(\()o Fn(~)-23 b(x)p Fp(\))p Fn( )1257 618 y Fj(+)1255 651 y Fl(x;\033)1308 639 y Fn( )1342 618 y Fi(\000)1340 651 y Fl(x;\033)1790 639 y Fp(\(27\))-13 780 y(with)20 b(\003)g(=)h(\()p Fm(\000)p Fn(\014)s(=)p Fp(2)p Fn(;)8 b(\014)s(=)p Fp(2\))14 b Fm(\002)f Fp(\()p Fm(\000)p Fn(N)q(=)p Fp(2)p Fn(;)8 b(N)q(=)p Fp(2\))22 b(and)f Fn(c)p Fp(\()o Fn(~)-23 b(x)o Fp(\))21 b(=)g Fn(c)p Fp(\()o Fn(~)-23 b(x)13 b Fp(+)h Fn(a)p Fp(\).)33 b(W)l(e)20 b(start)h(b)o(y)e(studying)i(the)-13 858 y(partition)16 b(function)g Fm(N)24 b Fg(i.e.)17 b Fp(the)f(denominator)g(of)g(eq.\(26\).)21 b(It)16 b(is)g(con)o(v)o (enien)o(t)e(to)j(rewrite)e(it)h(as:)580 986 y Fm(N)21 b Fp(=)694 927 y Fk(Z)744 986 y Fn(P)7 b Fp(\()p Fn(d )858 993 y Fl(i:r)o(:)907 986 y Fp(\))p Fn(e)949 965 y Fi(\000)p Fl(V)1004 954 y Ff(0)1021 965 y Fj(\()p Fl( )1058 970 y Fe(i:r)o(:)1104 965 y Fj(\))1790 986 y Fp(\(28\))580 1095 y Fn(e)603 1074 y Fi(\000)p Fl(V)659 1063 y Ff(0)676 1074 y Fj(\()p Fl( )713 1079 y Fe(i:r)o(:)758 1074 y Fj(\))788 1095 y Fp(=)840 1036 y Fk(Z)890 1095 y Fn(P)g Fp(\()p Fn(d )1004 1102 y Fl(u:v)q(:)1064 1095 y Fp(\))p Fn(e)1106 1074 y Fi(\000)1138 1066 y Fj(\026)1133 1074 y Fl(V)h Fj(\()p Fl( )1199 1079 y Fe(i:r)o(:)1244 1074 y Fj(+)p Fl( )1294 1078 y Fe(u:v)q(:)1350 1074 y Fj(\))1790 1095 y Fp(\(29\))-13 1223 y(where)18 b Fn( )162 1230 y Fl(u:v)q(:)222 1223 y Fp(,)g Fn( )286 1230 y Fl(i:r)o(:)335 1223 y Fp(,)g(are)g(grassmanian)h(\014elds)f(and)h Fn(P)7 b Fp(\()p Fn(d )1070 1230 y Fl(u:v)q(:)1130 1223 y Fp(\),)19 b Fn(P)7 b Fp(\()p Fn(d )1296 1230 y Fl(i:r)o(:)1345 1223 y Fp(\))18 b(denote)g(resp)q(ectiv)o(ely)e(the)-13 1301 y(grassmanian)g(in)o(tegrations)f(with)h(v)m(anishing)g(cross)g (propagator)i(and)e(with)g(propagators)i Fn(g)1722 1308 y Fl(u:v)q(:)1783 1301 y Fn(;)8 b(g)1828 1308 y Fl(i:r)o(:)-13 1379 y Fp(giv)o(en)15 b(b)o(y:)323 1520 y Fn(g)346 1527 y Fl(u:v)397 1520 y Fp(\()p Fn(x;)8 b(y)r Fp(\))41 b(=)631 1461 y Fk(Z)681 1520 y Fn(dk)809 1486 y(e)832 1468 y Fi(\000)p Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))p 738 1508 318 2 v 738 1562 a Fm(\000)p Fn(ik)819 1569 y Fj(0)850 1562 y Fp(+)896 1549 y Fn(~)899 1562 y(k)926 1547 y Fj(2)957 1562 y Fm(\000)10 b Fn(\031)1036 1547 y Fj(2)1061 1520 y Fp(\(1)h Fm(\000)g Fn(h)p Fp(\()p Fn(k)1239 1499 y Fj(2)1237 1532 y(0)1270 1520 y Fp(+)g(\()p Fm(j)1349 1507 y Fn(~)1352 1520 y(k)r Fm(j)f(\000)h Fn(\031)r Fp(\))1502 1499 y Fj(2)1521 1520 y Fp(\))250 b(\(30\))333 1663 y Fn(g)356 1670 y Fl(i:r)397 1663 y Fp(\()p Fn(x;)8 b(y)r Fp(\))41 b(=)631 1604 y Fk(Z)681 1663 y Fn(dk)809 1629 y(e)832 1611 y Fi(\000)p Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))p 738 1651 V 738 1705 a Fm(\000)p Fn(ik)819 1712 y Fj(0)850 1705 y Fp(+)896 1691 y Fn(~)899 1705 y(k)926 1690 y Fj(2)957 1705 y Fm(\000)10 b Fn(\031)1036 1690 y Fj(2)1061 1663 y Fn(h)p Fp(\()p Fn(k)1135 1642 y Fj(2)1133 1675 y(0)1165 1663 y Fp(+)h(\()p Fm(j)1244 1649 y Fn(~)1247 1663 y(k)r Fm(j)g(\000)g Fn(\031)r Fp(\))1398 1642 y Fj(2)1417 1663 y Fp(\))-13 1805 y(where)17 b Fn(h)p Fp(\()p Fn(t)p Fp(\))g(is)g(a)h Fn(C)361 1787 y Fi(1)415 1805 y Fp(function)f(in)g(its)g(argumen)o(t)f Fn(t)h Fp(whic)o(h)g(is)g(iden)o(tically)e(1)i(if)g Fn(t)e(>)h(\031)1620 1787 y Fj(2)1657 1805 y Fp(and)h(0)h(in)f(a)-13 1884 y(neigh)o(b)q(ourho)q(o)q(d)h(of)f(the)f(origin.)10 1962 y(In)g(\(B.G.P)l(.S.\))f(it)h(is)g(sho)o(wn)h(that)f(the)g(ultra)o (violet)f(part)i(of)f(the)g(propagator)j(can)d(b)q(e)g(written)g(as)361 2128 y Fn(g)384 2135 y Fl(u:v)q(:)444 2128 y Fp(\()p Fn(x)p Fp(\))e(=)g Fn(G)p Fp(\()p Fn(x)p Fp(\))d(+)g Fn(R)p Fp(\()p Fn(x)p Fp(\))98 b Fn(G)p Fp(\()p Fn(x)p Fp(\))14 b(=)f Fn(H)t Fp(\()o Fn(~)-23 b(x)p Fp(\))p Fn(H)t Fp(\()p Fn(x)1311 2135 y Fj(0)1331 2128 y Fp(\))1370 2094 y Fn(e)1393 2068 y Fi(\000)1433 2055 y Fe(~)-16 b(x)1450 2044 y Ff(2)p 1425 2061 50 2 v 1425 2082 a(2)p Fe(x)1457 2089 y Ff(0)p 1355 2116 143 2 v 1355 2125 a Fm(p)p 1396 2125 102 2 v 1396 2163 a Fp(2)p Fn(\031)r(x)1478 2170 y Fj(0)1790 2128 y Fp(\(31\))-13 2273 y(where)14 b Fn(H)t Fp(\()p Fn(t)p Fp(\))h(is)g(a)g(smo)q(oth)g(function)g(of)g (compact)f(supp)q(ort)i(suc)o(h)f(that)h Fn(H)t Fp(\()p Fn(t)p Fp(\))d(=)h Fn(e)1518 2255 y Fl(t)1531 2243 y Ff(2)1548 2255 y Fl(\031)1569 2243 y Ff(2)1603 2273 y Fp(if)h Fm(j)p Fn(t)p Fm(j)e(\024)h Fp(1)h(and)-13 2351 y Fn(H)t Fp(\()p Fn(t)p Fp(\))f(=)g(0)j(if)e Fm(j)p Fn(t)p Fm(j)f(\025)g Fn(\015)j(>)d Fp(1,)i(and)h Fn(R)p Fp(\()p Fn(x)p Fp(\))d Fm(\024)799 2331 y Fl(C)824 2337 y Fe(N)p 769 2340 114 2 v 769 2368 a Fj(1+)p Fi(j)p Fl(x)p Fi(j)854 2359 y Fe(N)904 2351 y Fp(for)j(an)o(y)f(in)o(teger)f Fn(N)5 b Fp(.)22 b(A)16 b(minor)f(adaptation)j(of)f(the)-13 2430 y(ultra)o(violet)f(problem)h(analysis)h(for)g(the)g Fn(u)e Fp(=)h(0)h(case)g(in)f(\(B.G.P)l(.S\))g(to)h(the)g Fn(u)e Fm(6)p Fp(=)h(0)h(case)g(allo)o(ws)g(to)-13 2508 y(c)o(hec)o(k)e(that)i(there)g(exists)f(an)i Fn(")f Fp(suc)o(h)f(that)i Fn(V)859 2490 y Fj(0)897 2508 y Fp(can)f(b)q(e)g(written,)g(for)g Fm(j)p Fn(z)r Fm(j)e(\024)h Fn(")p Fp(,)h(if)f Fn(z)i Fp(=)e(\()p Fn(\025;)8 b(\027;)g(u)p Fp(\),)18 b(in)-13 2586 y(the)e(follo)o(wing)g(w)o(a)o(y:)907 2728 y(12)p eop %%Page: 13 13 13 12 bop 113 105 a Fn(V)152 85 y Fj(0)172 105 y Fp(\()p Fn( )r Fp(\))13 b(=)310 64 y Fk(X)309 156 y Fl(\033)o(;\033)359 147 y Fh(0)379 47 y Fk(Z)429 105 y Fn(dk)479 112 y Fj(1)499 105 y Fn(dk)549 112 y Fj(2)570 105 y Fn(dk)620 112 y Fj(3)640 105 y Fn(dk)690 112 y Fj(4)712 105 y Fp(^)-25 b Fn(v)q Fp(\()752 92 y Fn(~)755 105 y(k)780 112 y Fj(1)811 105 y Fm(\000)858 92 y Fn(~)861 105 y(k)886 112 y Fj(4)906 105 y Fp(\))p Fn( )959 85 y Fj(+)957 118 y Fl(k)975 123 y Ff(1)992 118 y Fl(;\033)1026 105 y Fn( )1060 85 y Fj(+)1058 118 y Fl(k)1076 123 y Ff(2)1093 118 y Fl(;\033)1124 109 y Fh(0)1137 105 y Fn( )1171 85 y Fi(\000)1169 118 y Fl(k)1187 123 y Ff(3)1205 118 y Fl(;\033)1236 109 y Fh(0)1249 105 y Fn( )1283 85 y Fi(\000)1281 118 y Fl(k)1299 123 y Ff(4)1317 118 y Fl(;\033)1350 105 y Fn(\016)r Fp(\()p Fn(k)1418 112 y Fj(1)1448 105 y Fp(+)11 b Fn(k)1522 112 y Fj(2)1554 105 y Fm(\000)f Fn(k)1628 112 y Fj(3)1659 105 y Fm(\000)h Fn(k)1734 112 y Fj(4)1754 105 y Fp(\))g(+)113 242 y(+2)183 200 y Fk(X)203 288 y Fl(\033)252 242 y Fn(\025)288 183 y Fk(Z)338 242 y Fn(dk)r(dp)r Fp(^)-26 b Fn(v)s Fp(\()p Fn(~)i(p)p Fp(\))539 229 y(^)528 242 y Fn(R)q Fp(\()p Fn(~)g(p)11 b Fm(\000)668 229 y Fn(~)670 242 y(k)r Fp(\))p Fn( )750 221 y Fj(+)748 255 y Fl(k)q(;\033)800 242 y Fn( )834 221 y Fi(\000)832 255 y Fl(k)q(;\033)896 242 y Fp(+)g(\()p Fn(\027)j Fm(\000)d Fp(4)p Fn(\031)r(\025)r Fp(^)-26 b Fn(v)r Fp(\(0\))1232 229 y(^)1222 242 y Fn(R)p Fp(\(0\)\))1348 200 y Fk(X)1368 288 y Fl(\033)1417 183 y Fk(Z)1467 242 y Fn(dk)r( )1553 221 y Fj(+)1551 255 y Fl(k)q(;\033)1604 242 y Fn( )1638 221 y Fi(\000)1636 255 y Fl(k)q(;\033)1699 242 y Fp(+)113 381 y(+)159 339 y Fk(X)179 426 y Fl(\033)243 327 y Fi(1)231 339 y Fk(X)227 430 y Fl(n)p Fj(=1)302 381 y Fn(uc)351 388 y Fl(n)383 322 y Fk(Z)433 381 y Fn(dk)r Fp([)p Fn( )533 360 y Fj(+)531 393 y Fl(k)q Fj(+)p Fl(n\031)q(;\033)653 381 y Fn( )687 360 y Fi(\000)685 393 y Fl(k)q Fi(\000)p Fl(n\031)q(;\033)819 381 y Fp(+)11 b Fn( )902 360 y Fj(+)900 393 y Fl(k)q Fi(\000)p Fl(n\031)q(;\033)1022 381 y Fn( )1056 360 y Fi(\000)1054 393 y Fl(k)q Fj(+)p Fl(n\031)q(;\033)1177 381 y Fp(])f(+)551 b(\(32\))113 523 y(+)291 482 y Fk(X)159 576 y Fj(\()p Fl(\033)193 581 y Ff(1)210 576 y Fl(;k)238 581 y Ff(1)256 576 y Fj(\))p Fl(;:::)o(;)p Fj(\()p Fl(\033)353 581 y Ff(2)p Fe(m)397 576 y Fl(;k)425 581 y Ff(2)p Fe(m)470 576 y Fj(\))507 469 y Fi(1)495 482 y Fk(X)492 572 y Fl(n)p Fj(=0)567 464 y Fk(Z)616 523 y Fn(dk)666 530 y Fj(1)695 523 y Fn(:)8 b(:)g(:)g(dk)811 530 y Fj(2)p Fl(m)862 523 y Fn(W)908 530 y Fl(m;n)973 523 y Fp(\()p Fn(k)1017 530 y Fj(1)1037 523 y Fn(;)g(:)g(:)g(:)f(;)h(k)1171 530 y Fj(2)p Fl(m)1222 523 y Fp(;)g Fn(z)r Fp(\))p Fn( )1322 502 y Fj(+)1320 536 y Fl(k)1338 541 y Ff(1)1356 536 y Fl(;\033)1386 541 y Ff(1)1413 523 y Fn(:)g(:)g(:)g( )1513 502 y Fj(+)1511 536 y Fl(k)1529 540 y Fe(m)1558 536 y Fl(;\033)1588 540 y Fe(m)1631 523 y Fm(\001)113 645 y Fn( )147 625 y Fi(\000)145 658 y Fl(k)163 663 y Fe(m)p Ff(+1)231 658 y Fl(;\033)261 663 y Fe(m)p Ff(+1)339 645 y Fn(:)g(:)g(:)g( )439 625 y Fi(\000)437 658 y Fl(k)455 663 y Ff(2)p Fe(m)499 658 y Fl(;\033)529 663 y Ff(2)p Fe(m)576 645 y Fn(\016)r Fp(\()p Fn(k)644 652 y Fj(1)674 645 y Fp(+)j Fn(:)d(:)g(:)j Fp(+)g Fn(k)866 652 y Fl(m)910 645 y Fm(\000)g Fn(k)985 652 y Fl(m)p Fj(+1)1075 645 y Fm(\000)g Fn(:)d(:)g(:)i Fm(\000)h Fn(k)1268 652 y Fj(2)p Fl(m)1330 645 y Fp(+)g(2)p Fn(n\031)r Fp(\))-13 755 y(where)16 b Fn(c)p Fp(\()o Fn(~)-23 b(x)o Fp(\))14 b(=)280 722 y Fk(P)324 735 y Fi(1)324 768 y Fl(n)p Fj(=0)401 755 y Fn(c)422 762 y Fl(n)453 755 y Fp(cos)q(\(2)p Fn(n\031)q(~)-23 b(x)o Fp(\),)16 b Fn(n\031)i Fp(is)e(a)h(spatial)f(v)o(ector)f(and)i (the)f(k)o(ernels)-13 834 y Fn(W)33 841 y Fl(m;n)97 834 y Fp(\()p Fn(k)141 841 y Fj(1)161 834 y Fn(;)8 b(:)g(:)g(:)g(;)g(k)296 841 y Fl(m)329 834 y Fp(;)g Fn(z)r Fp(\))16 b(are)h Fn(C)532 815 y Fi(1)585 834 y Fp(b)q(ounded)g(functions)f(suc)o(h)g(that)h Fn(W)1256 841 y Fl(m;n)1334 834 y Fp(=)d Fn(W)1432 841 y Fl(m;)p Fi(\000)p Fl(n)1540 834 y Fp(and)103 956 y(sup)176 968 y Fl(k)194 973 y Ff(1)212 968 y Fl(;:::)o(;k)279 973 y Ff(2)p Fe(m)326 956 y Fm(j)p Fn(W)386 963 y Fl(m;n)450 956 y Fp(\()p Fn(k)494 963 y Fj(1)514 956 y Fn(;)8 b(:)g(:)g(:)g(;)g(k) 649 963 y Fj(2)p Fl(m)700 956 y Fp(;)g Fn(z)r Fp(\))p Fm(j)13 b(\024)h Fn(C)885 935 y Fl(m)918 956 y Fn(z)943 935 y Fj(max\(2)p Fl(;)1057 922 y Ff(m)p 1054 928 25 2 v 1059 948 a(2)1084 935 y Fi(\000)p Fj(1\))1237 902 y Fi(1)1224 914 y Fk(X)1227 1001 y Fe(k)q Ff(=0)1158 1029 y Fe(n)1177 1036 y Ff(1)1195 1029 y(+)p Fe(:::)r Ff(+)p Fe(n)1289 1038 y(k)1308 1029 y Ff(=)p Fe(n)1364 956 y Fm(j)p Fn(uc)1427 963 y Fl(n)1448 968 y Ff(1)1467 956 y Fm(j)8 b Fn(:)g(:)g(:)g Fm(j)p Fn(uc)1618 963 y Fl(n)1639 969 y Fe(k)1660 956 y Fm(j)116 b Fp(\(33\))-13 1126 y(where)14 b(the)g(sup)g(is)g(o)o(v)o(er)g(the)g(momen)o(ta)d(in)j (the)g(supp)q(ort)i(of)e Fn(h)p Fp(\()p Fn(k)1180 1108 y Fj(2)1178 1139 y(0)1207 1126 y Fp(+)7 b(\()p Fm(j)1282 1113 y Fn(~)1285 1126 y(k)r Fm(j)g(\000)g Fn(\031)r Fp(\))1428 1108 y Fj(2)1447 1126 y Fp(\))14 b(and)h Fn(C)i Fp(is)d(a)h(suitable) -13 1205 y(constan)o(t)i(and)f(of)h(course)193 1273 y Fk(X)192 1360 y Fl(m;n)263 1314 y Fp(sup)336 1326 y Fl(k)354 1331 y Ff(1)372 1326 y Fl(;:::)o(;k)439 1331 y Ff(2)p Fe(m)486 1314 y Fm(j)p Fn(W)546 1321 y Fl(m;n)610 1314 y Fp(\()p Fn(k)654 1321 y Fj(1)674 1314 y Fn(;)8 b(:)g(:)g(:)g(;)g(k) 809 1321 y Fj(2)p Fl(m)860 1314 y Fp(;)g Fn(z)r Fp(\))p Fm(j)13 b(\024)1018 1260 y Fi(1)1006 1273 y Fk(X)1006 1360 y Fl(m)p Fj(=)1074 1314 y Fn(C)1113 1294 y Fl(m)1146 1314 y Fn(z)1171 1294 y Fj(max\(2)p Fl(;)1285 1280 y Ff(m)p 1282 1286 V 1287 1307 a(2)1312 1294 y Fi(\000)p Fj(1\))1396 1260 y Fi(1)1384 1273 y Fk(X)1381 1365 y Fl(k)q Fj(=0)1446 1314 y Fp(\()1480 1260 y Fi(1)1468 1273 y Fk(X)1465 1364 y Fl(n)p Fj(=0)1540 1314 y Fn(u)p Fm(j)p Fn(c)1603 1321 y Fl(n)1626 1314 y Fm(j)p Fp(\))1659 1294 y Fl(k)1790 1314 y Fp(\(34\))-13 1441 y(con)o(v)o(erges)20 b(for)h(small)e(enough)j Fn(z)r Fp(.)35 b(W)l(e)21 b(shall)g(not)g (repro)q(duce)g(here)f(the)h(details)f(of)i(the)e(pro)q(of)i(of)-13 1519 y(eq.\(32\))16 b(as)h(this)f(w)o(ould)g(b)q(e)g(a)h(w)o(ord)g(b)o (y)e(w)o(ord)i(rep)q(etition)e(of)i(par.)22 b(3)16 b(of)h(\(B.G.P)l (.S.\).)10 1598 y(In)d(order)f(to)h(p)q(erform)f(the)h(\\infrared")g (in)o(tegration)f(eq.\(28\))g(w)o(e)g(decomp)q(ose)g(the)h(grassmanian) g(in-)-13 1676 y(tegration)d Fn(P)c Fp(\()p Fn(d )r Fp(\))k(in)o(to)f (a)i(pro)q(duct)f(of)g(indep)q(enden)o(t)f(grassmanian)h(in)o (tegrations,)h(that)f(is)g Fn(P)c Fp(\()p Fn(d )1772 1683 y Fl(i:r)o(:)1821 1676 y Fp(\))13 b(=)-13 1721 y Fk(Q)26 1734 y Fj(0)26 1766 y Fl(h)p Fj(=)p Fi(\0001)147 1754 y Fn(P)7 b Fp(\()p Fn(d )263 1736 y Fl(h)285 1754 y Fp(\).)20 b(This)14 b(can)f(b)q(e)g(done)g(b)o(y)g(setting)g Fn(g)954 1761 y Fl(i:r)o(:)1003 1754 y Fp(\()p Fn(k)r Fp(\))g(=)1133 1721 y Fk(P)1177 1734 y Fj(0)1177 1766 y Fl(h)p Fj(=)p Fi(\0001)1298 1754 y Fn(g)1323 1736 y Fl(h)1346 1754 y Fp(\()p Fn(k)r Fp(\))g(and)g(b)o(y)g(writing)f Fn( )1778 1734 y Fi(\006)1776 1766 y Fl(i:r)o(:)1839 1754 y Fp(=)-13 1799 y Fk(P)31 1843 y Fl(h)61 1832 y Fn( )95 1814 y Fi(\006)p Fl(;h)154 1832 y Fp(,)23 b(with)f Fn( )342 1814 y Fl(h)386 1832 y Fp(b)q(eing)g(a)h(family)c(of)k (grassmanian)f(\014elds)g(with)f(v)m(anishing)i(\\cross)g(propaga-)-13 1911 y(tor")14 b(\()p Fg(i.e.)k Fp(an)d(indep)q(enden)o(t)e(family)e (of)k(v)m(ariables\))f(and)g(with)g(propagator)1398 1875 y Fk(R)1434 1911 y Fn( )1468 1893 y Fl(h)1466 1923 y(k)1484 1928 y Ff(1)1501 1923 y Fl(;\033)1531 1928 y Ff(1)1551 1911 y Fn( )1585 1893 y Fl(h)1583 1923 y(k)1601 1928 y Ff(2)1618 1923 y Fl(;\033)1648 1928 y Ff(2)1667 1911 y Fn(P)7 b Fp(\()p Fn(d )1783 1893 y Fl(h)1806 1911 y Fp(\))14 b(=)-13 1989 y Fn(\016)9 1996 y Fl(\033)29 2001 y Ff(1)46 1996 y Fl(;\033)76 2001 y Ff(2)95 1989 y Fn(\016)r Fp(\()p Fn(k)163 1996 y Fj(1)194 1989 y Fm(\000)d Fn(k)269 1996 y Fj(2)289 1989 y Fp(\))p Fn(g)333 1971 y Fl(h)355 1989 y Fp(\()p Fn(k)399 1996 y Fj(1)419 1989 y Fp(\):)582 2089 y Fn(g)607 2069 y Fl(h)630 2089 y Fp(\()p Fn(k)r Fp(\))j(=)766 2056 y Fn(f)5 b Fp(\()p Fn(\015)842 2038 y Fi(\000)p Fj(2)p Fl(h)909 2056 y Fp(\()p Fn(k)955 2038 y Fj(2)953 2068 y(0)986 2056 y Fp(+)11 b(\()p Fm(j)1065 2042 y Fn(~)1068 2056 y(k)r Fm(j)g(\000)g Fn(\031)r Fp(\))1219 2038 y Fj(2)1238 2056 y Fp(\)\))p 766 2078 511 2 v 888 2131 a Fm(\000)p Fn(ik)969 2138 y Fj(0)1000 2131 y Fp(+)g Fn(E)s Fp(\()1104 2118 y Fn(~)1107 2131 y(k)r Fp(\))1790 2089 y(\(35\))-13 2216 y(where)g Fn(E)s Fp(\()178 2203 y Fn(~)181 2216 y(k)r Fp(\))j(=)290 2203 y Fn(~)293 2216 y(k)320 2198 y Fj(2)350 2216 y Fm(\000)d Fn(\031)430 2198 y Fj(2)449 2216 y Fp(,)h Fn(f)5 b Fp(\()p Fn(\015)551 2198 y Fi(\000)p Fj(2)p Fl(h)619 2216 y Fn(t)637 2198 y Fj(2)657 2216 y Fp(\))13 b(=)h Fn(h)p Fp(\()p Fn(\015)816 2198 y Fi(\000)p Fj(2)p Fl(h)p Fi(\000)p Fj(2)929 2216 y Fn(t)947 2198 y Fj(2)966 2216 y Fp(\))q Fm(\000)q Fn(h)p Fp(\()p Fn(\015)1101 2198 y Fi(\000)p Fj(2)p Fl(h)1169 2216 y Fn(t)1187 2198 y Fj(2)1207 2216 y Fp(\))d(is)h(a)g Fn(C)1357 2198 y Fi(1)1405 2216 y Fp(function)f(with)h(compact)-13 2294 y(supp)q(ort.)10 2372 y(It)22 b(is)f(con)o(v)o(enien)o(t)f(to)i (in)o(tro)q(duce)f(new)h(grassmanian)g(\014elds)g Fn( )1213 2349 y Fi(\006)p Fl(;h)1211 2386 y(k)q(;~)-18 b(!)q(;\033)1296 2372 y Fp(,)23 b(called)d Fg(quasi-p)n(articles)j Fp(Eu-)-13 2451 y(clidean)15 b(\014elds,)h(with)g(propagators)i Fn(g)695 2432 y Fl(h)693 2463 y(~)-18 b(!)719 2451 y Fp(\()p Fn(k)r Fp(\))16 b(and)h(v)m(anishing)g(cross)f(propagators,)i (suc)o(h)e(that:)520 2560 y Fn( )554 2537 y Fi(\006)p Fl(;h)552 2573 y(k)q(;\033)627 2560 y Fp(=)696 2519 y Fk(X)679 2612 y Fl(~)-19 b(!)q Fj(=)p Fi(\006)p Fj(1)782 2560 y Fn( )816 2537 y Fi(\006)p Fl(;h)814 2574 y(k)q(;~)h(!)q(;\033) 948 2560 y Fn(g)973 2540 y Fl(h)996 2560 y Fp(\()p Fn(k)r Fp(\))14 b(=)1144 2519 y Fk(X)1127 2612 y Fl(~)-19 b(!)q Fj(=)p Fi(\006)p Fj(1)1230 2560 y Fn(g)1255 2540 y Fl(h)1253 2573 y(~)h(!)1279 2560 y Fp(\()p Fn(k)r Fp(\))907 2728 y(13)p eop %%Page: 14 14 14 13 bop -13 -5 a Fp(with:)518 82 y Fn(g)543 61 y Fl(h)541 94 y(~)-18 b(!)567 82 y Fp(\()p Fn(k)r Fp(\))14 b(=)f Fn(\037)p Fp(\()p Fn(~)-24 b(!)776 69 y(~)779 82 y(k)r Fp(\))830 48 y Fn(f)5 b Fp(\()p Fn(\015)906 30 y Fi(\000)p Fj(2)p Fl(h)974 48 y Fp(\()p Fn(k)1020 30 y Fj(2)1018 60 y(0)1051 48 y Fp(+)11 b(\()p Fm(j)1130 35 y Fn(~)1133 48 y(k)q Fm(j)g(\000)g Fn(\031)r Fp(\))1283 30 y Fj(2)1302 48 y Fp(\)\))p 830 70 511 2 v 953 124 a Fm(\000)p Fn(ik)1034 131 y Fj(0)1064 124 y Fp(+)g Fn(E)s Fp(\()1168 111 y Fn(~)1171 124 y(k)r Fp(\))1790 82 y(\(36\))-13 210 y(where)18 b Fn(\037)p Fp(\()177 197 y Fn(~)180 210 y(k)q Fp(\))13 b(+)f Fn(\037)p Fp(\()p Fm(\000)374 197 y Fn(~)377 210 y(k)q Fp(\))17 b(=)h(1,)g Fn(\037)p Fp(\()598 197 y Fn(~)601 210 y(k)r Fp(\))f(=)g(1)i(if)806 197 y Fn(~)808 210 y(k)h Fm(\025)d Fp(0)h(and)h(it)f(is)g(0)g(otherwise.)27 b(It)18 b(will)f(b)q(e)i(con)o(v)o(enien)o(t)-13 288 y(in)f(the)g(follo)o(wing) g(to)g(write)g(the)g(momen)o(tum)c(of)19 b(the)f Fn(~)-24 b(!)20 b Fp(fermion)c(as)1305 275 y Fn(~)1307 288 y(k)f Fp(+)d Fn(~)-24 b(!)r(p)1453 295 y Fl(F)1483 288 y Fp(,)18 b(where)1656 275 y Fn(~)1658 288 y(k)j Fp(is)d(called)-13 366 y(the)d Fg(momentum)j(me)n(asur)n(e)n(d)d(fr)n(om)h(the)h(F)l(ermi) g(surfac)n(e)e Fp(and)i Fn(p)1155 373 y Fl(F)1198 366 y Fp(=)d Fn(\031)j Fp(is)f(the)f(F)l(ermi)e(momen)o(tum)o(.)18 b(It)-13 445 y(is)d(easy)g(to)g(c)o(hec)o(k,)e(see)h(\(B.G.P)l(.S.\),)f (that)j(the)e(quasi-particle)g(propagator)j(eq.\(36\))e(can)g(b)q(e)g (written)-13 523 y(as:)509 601 y Fn(g)534 581 y Fl(h)532 613 y(~)-18 b(!)558 601 y Fp(\()p Fn(k)13 b Fp(+)e Fn(~)-24 b(!)r(p)720 608 y Fl(F)750 601 y Fp(\))14 b(=)f Fn(\015)862 581 y Fi(\000)p Fl(h)912 601 y Fn(g)935 608 y Fl(~)-18 b(!)961 601 y Fp(\()p Fn(\015)1008 581 y Fi(\000)p Fl(h)1058 601 y Fn(k)r Fp(\))11 b(+)h(\026)-25 b Fn(g)1189 581 y Fl(h)1211 601 y Fp(\()p Fn(\015)1258 581 y Fi(\000)p Fl(h)1308 601 y Fn(k)r Fp(\))-13 700 y(with)18 b(\026)-26 b Fn(g)123 682 y Fl(h)145 700 y Fp(\()p Fn(k)r Fp(\))17 b(regular)f(and)h(w)o(eakly)e(dep)q(enden)o(t)h(on)h Fn(h)f Fp(and:)727 835 y Fn(g)750 842 y Fl(~)-18 b(!)775 835 y Fp(\()p Fn(k)r Fp(\))14 b(=)911 801 y Fn(f)5 b Fp(\()p Fn(k)986 783 y Fj(2)984 813 y(0)1017 801 y Fp(+)1063 788 y Fn(~)1066 801 y(k)1093 783 y Fj(2)1113 801 y Fp(\))p 911 823 221 2 v 912 877 a Fm(\000)p Fn(ik)993 884 y Fj(0)1023 877 y Fp(+)11 b Fn(~)-24 b(!)1102 863 y(~)1104 877 y(k)-13 970 y Fp(W)l(e)23 b(can)g(sa)o(y)g(that)g Fn( )411 947 y Fl(h;)p Fi(\006)409 984 y Fl(k)q Fj(+)p Fl(~)-18 b(!)r(p)497 990 y Fe(F)522 984 y Fl(;~)g(!)q(;\033)611 970 y Fp(ha)o(v)o(e)22 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Fn(dk)441 1863 y Fl(i)456 1856 y Fn( )490 1832 y Fj(+)p Fl(;)p Fi(\024)p Fj(0)488 1869 y Fl(k)506 1874 y Ff(1)523 1869 y Fj(+)p Fl(~)-18 b(!)572 1874 y Ff(1)590 1869 y Fl(p)608 1875 y Fe(F)633 1869 y Fl(;~)g(!)665 1874 y Ff(1)682 1869 y Fl(;\033)715 1856 y Fn( )749 1832 y Fj(+)p Fl(;)p Fi(\024)p Fj(0)747 1869 y Fl(k)765 1874 y Ff(2)783 1869 y Fj(+)p Fl(~)g(!)832 1874 y Ff(2)849 1869 y Fl(p)867 1875 y Fe(F)893 1869 y Fl(;~)g(!)925 1874 y Ff(2)942 1869 y Fl(;\033)973 1860 y Fh(0)986 1856 y Fn( )1020 1832 y Fi(\000)p Fl(;)p Fi(\024)p Fj(0)1018 1869 y Fl(k)1036 1874 y Ff(3)1053 1869 y Fj(+)p Fl(~)g(!)1102 1874 y Ff(3)1120 1869 y Fl(p)1138 1875 y Fe(F)1163 1869 y Fl(;~)g(!)1195 1874 y Ff(3)1212 1869 y Fl(;\033)1243 1860 y Fh(0)1257 1856 y Fn( )1291 1832 y Fi(\000)p Fl(;)p Fi(\024)p Fj(0)1289 1869 y Fl(k)1307 1874 y Ff(4)1324 1869 y Fj(+)p Fl(~)g(!)1373 1874 y Ff(4)1391 1869 y Fl(p)1409 1875 y Fe(F)1434 1869 y Fl(;~)g(!)1466 1874 y Ff(4)1483 1869 y Fl(;\033)1516 1856 y Fn(\016)r Fp(\()1559 1815 y Fk(X)1583 1906 y Fl(i)1627 1856 y Fn(")1650 1863 y Fl(i)1664 1856 y Fn(k)1689 1863 y Fl(i)1703 1856 y Fp(\))-13 1998 y(where)16 b Fn(")151 2005 y Fj(1)184 1998 y Fp(=)e Fn(")259 2005 y Fj(2)292 1998 y Fp(=)g Fm(\000)p Fn(")406 2005 y Fj(3)439 1998 y Fp(=)g Fm(\000)p Fn(")553 2005 y Fj(4)586 1998 y Fp(=)g(1)i(and)h(w)o(e)f(call)310 2126 y Fn(F)349 2104 y Fi(\024)p Fj(0)342 2137 y(1)409 2126 y Fp(=)461 2085 y Fk(X)480 2178 y Fl(~)-19 b(!)529 2126 y Fn(F)568 2106 y Fi(\024)p Fj(0)615 2126 y Fp(\()p Fn(~)-24 b(!)r(;)8 b Fm(\000)p Fn(~)-24 b(!)r(;)8 b(~)-24 b(!)r(;)8 b Fm(\000)p Fn(~)-24 b(!)q Fp(\))98 b Fn(F)1061 2104 y Fi(\024)p Fj(0)1054 2137 y(2)1121 2126 y Fp(=)1173 2085 y Fk(X)1192 2178 y Fl(~)-18 b(!)1241 2126 y Fn(F)1280 2106 y Fi(\024)p Fj(0)1327 2126 y Fp(\()p Fn(~)-24 b(!)r(;)8 b Fm(\000)p Fn(~)-24 b(!)r(;)8 b Fm(\000)p Fn(~)-24 b(!)q(;)8 b(~)-24 b(!)s Fp(\))310 2244 y Fn(F)349 2222 y Fi(\024)p Fj(0)342 2255 y(3)409 2244 y Fp(=)461 2203 y Fk(X)480 2296 y Fl(~)-19 b(!)529 2244 y Fn(F)568 2224 y Fi(\024)p Fj(0)615 2244 y Fp(\()p Fn(~)-24 b(!)r(;)8 b(~)-24 b(!)r(;)8 b Fm(\000)p Fn(~)-24 b(!)r(;)8 b Fm(\000)p Fn(~)-24 b(!)q Fp(\))98 b Fn(F)1061 2222 y Fi(\024)p Fj(0)1054 2255 y(4)1121 2244 y Fp(=)1173 2203 y Fk(X)1192 2296 y Fl(~)-18 b(!)1241 2244 y Fn(F)7 b Fp(\()p Fn(~)-24 b(!)r(;)8 b(~)-24 b(!)r(;)8 b(~)-24 b(!)r(;)8 b(~)-24 b(!)r Fp(\))278 b(\(47\))310 2362 y Fn(F)349 2340 y Fi(\024)p Fj(0)342 2373 y(5)409 2362 y Fp(=)461 2321 y Fk(X)480 2414 y Fl(~)-19 b(!)529 2362 y Fn(F)568 2342 y Fi(\024)p Fj(0)615 2362 y Fp(\()p Fn(~)-24 b(!)r(;)8 b(~)-24 b(!)r(;)8 b(~)-24 b(!)r(;)8 b Fm(\000)p Fn(~)-24 b(!)r Fp(\))97 b Fn(F)1022 2340 y Fi(\024)p Fj(0)1015 2373 y(6)1083 2362 y Fp(=)1134 2321 y Fk(X)1153 2414 y Fl(~)-18 b(!)1203 2362 y Fn(F)1242 2342 y Fi(\024)p Fj(0)1288 2362 y Fp(\()p Fn(~)-24 b(!)r(;)8 b Fm(\000)p Fn(~)-24 b(!)r(;)8 b(~)-24 b(!)r(;)8 b(~)-24 b(!)r Fp(\))5 2509 y Fd(Remark:)22 b Fp(w)o(e)d(can)h(in)o(terpret)e (eq.\(46\))h(as)i(stating)f(that)g(the)f(relev)m(an)o(t)g(part)g(of)h (the)g(in)o(teraction)109 2587 y(b)q(et)o(w)o(een)12 b(fermions)g(is)h(the)g(one)g(whic)o(h)g(in)o(v)o(olv)o(es)e(fermions)h (with)h(momen)o(ta)d Fn(k)d Fp(+)e Fn(~)-24 b(!)r(p)1660 2594 y Fl(F)1703 2587 y Fp(near)14 b(the)907 2728 y(16)p eop %%Page: 17 17 17 16 bop 109 -5 a Fp(F)l(ermi)10 b(surface.)20 b(Note)13 b(moreo)o(v)o(er)e(that)i(the)g(p)q(erio)q(dic)f(p)q(oten)o(tial)h(has) g(the)g(e\013ect)g(that)g(the)g(sum)109 74 y(of)j(the)g(momen)o(ta)e (is)i(not)g(0)h(but)f(it)g(is)g(equal)f(to)i(2)p Fn(n\031)r Fp(;)e(this)h(phenomenon)g(is)g(called)f Fg(Umklapp)109 152 y Fp(and)h(w)o(e)f(call)g(the)g(terms)f(in)i(eq.\(46\))f(in)g(whic) o(h)g(the)g(momen)o(tum)d(is)j(not)h(conserv)o(ed)f Fg(Umklapp)109 230 y(terms)p Fp(.)31 b(It)20 b(will)e(b)q(e)i(crucial)f(in)g(the)h (follo)o(wing)f(the)h(\(trivial\))e(observ)m(ation)j(that)f Fn(\033)1687 237 y Fj(0)1720 230 y Fp(=)14 b Fn(O)q Fp(\()p Fn(u)p Fp(\))p 1659 249 218 2 v 109 308 a(and)j Fn(i)221 315 y Fj(0)240 308 y Fn(;)8 b(t)280 315 y Fj(0)299 308 y Fn(;)g(\025)349 315 y Fj(3)p Fl(;)p Fj(0)397 308 y Fn(;)g(\025)447 315 y Fj(5)p Fl(;)p Fj(0)494 308 y Fn(;)g(\025)544 315 y Fj(6)p Fl(;)p Fj(0)605 308 y Fp(=)14 b Fn(O)q Fp(\()p Fn(\025u)p Fp(\))p 204 329 587 2 v 1 w(.)-13 424 y(In)i(the)g(spinless) g(case)g Fn(\033)f Fp(=)f(0)j(and)228 547 y Fm(L)p Fn(V)290 554 y Fj(0)324 547 y Fp(=)d Fn(n)405 554 y Fj(0)425 547 y Fn(F)464 526 y Fi(\024)p Fj(0)457 559 y Fl(\027)521 547 y Fp(+)d Fn(a)596 554 y Fj(0)615 547 y Fn(F)654 526 y Fi(\024)p Fj(0)647 559 y Fl(\013)712 547 y Fp(+)g Fn(z)784 554 y Fj(0)803 547 y Fn(F)842 524 y Fi(\024)p Fj(0)835 559 y Fl(\020)900 547 y Fp(+)g Fn(s)972 554 y Fj(0)992 547 y Fn(F)1031 526 y Fi(\024)p Fj(0)1024 559 y Fl(\033)1088 547 y Fp(+)g Fn(i)1154 554 y Fj(0)1174 547 y Fn(F)1213 524 y Fi(\024)p Fj(0)1206 558 y Fl(i)1270 547 y Fp(+)g Fn(t)1337 554 y Fj(0)1357 547 y Fn(F)1396 526 y Fi(\024)p Fj(0)1389 559 y Fl(\034)1453 547 y Fp(+)g Fn(\025)1530 554 y Fj(0)1550 547 y Fn(F)1589 526 y Fi(\024)p Fj(0)1790 547 y Fp(\(48\))-13 670 y(where:)209 799 y Fn(F)248 778 y Fi(\024)p Fj(0)309 799 y Fp(=)360 757 y Fk(X)379 850 y Fl(~)-18 b(!)429 740 y Fk(Z)498 745 y Fj(4)480 757 y Fk(Y)478 848 y Fl(i)p Fj(=1)544 799 y Fn(dk)594 806 y Fl(i)609 799 y Fn( )643 775 y Fj(+)p Fl(;)p Fi(\024)p Fj(0)641 812 y Fl(k)659 817 y Ff(1)676 812 y Fj(+)p Fl(~)g(!)r(p)745 818 y Fe(F)770 812 y Fl(;~)g(!)805 799 y Fn( )839 775 y Fj(+)p Fl(;)p Fi(\024)p Fj(0)837 812 y Fl(k)855 817 y Ff(2)872 812 y Fi(\000)p Fl(~)g(!)r(p)941 818 y Fe(F)966 812 y Fl(;)p Fi(\000)p Fl(~)g(!)1028 799 y Fn( )1062 775 y Fi(\000)p Fl(;)p Fi(\024)p Fj(0)1060 812 y Fl(k)1078 817 y Ff(3)1096 812 y Fi(\000)p Fl(~)g(!)q(p)1164 818 y Fe(F)1189 812 y Fl(;)p Fi(\000)p Fl(~)g(!)1252 799 y Fn( )1286 775 y Fi(\000)p Fl(;)p Fi(\024)p Fj(0)1284 812 y Fl(k)1302 817 y Ff(4)1319 812 y Fj(+)p Fl(~)g(!)r(p)1388 818 y Fe(F)1413 812 y Fl(;~)g(!)1448 799 y Fn(\016)r Fp(\()1491 757 y Fk(X)1514 848 y Fl(i)1559 799 y Fn(")1582 806 y Fl(i)1596 799 y Fn(k)1621 806 y Fl(i)1635 799 y Fp(\))136 b(\(49\))-13 948 y(as,)16 b(in)g(the)h(spinless)f(case,)g Fn(F)535 925 y Fi(\024)p Fj(0)528 959 y(1)595 948 y Fp(=)e Fn(F)686 925 y Fi(\024)p Fj(0)679 959 y(2)747 948 y Fp(=)g Fn(F)838 930 y Fi(\024)p Fj(0)901 948 y Fp(and)j Fn(F)1035 925 y Fi(\024)p Fj(0)1028 959 y(3)1095 948 y Fp(=)d Fn(F)1186 925 y Fi(\024)p Fj(0)1179 959 y(4)1247 948 y Fp(=)g Fn(F)1338 925 y Fi(\024)p Fj(0)1331 959 y(5)1399 948 y Fp(=)g Fn(F)1490 925 y Fi(\024)p Fj(0)1483 959 y(6)1550 948 y Fp(=)g(0,)j(b)q(ecause)f (of)-13 1026 y(the)g(an)o(ticomm)n(utation)e(prop)q(erties)i(of)h(the)f (grassmanian)h(v)m(ariables.)10 1105 y(W)l(e)f(p)q(erform)g(eac)o(h)f (in)o(tegration)h(in)g(eq.\(39\))g(b)o(y)g(writing)556 1228 y Fn(V)595 1207 y Fl(h)618 1228 y Fp(\()p Fn( )671 1207 y Fi(\024)p Fl(h)720 1228 y Fp(\))e(=)g Fm(L)p Fn(V)878 1207 y Fl(h)901 1228 y Fp(\()p Fn( )954 1207 y Fi(\024)p Fl(h)1003 1228 y Fp(\))d(+)g Fm(R)p Fn(V)1164 1207 y Fl(h)1186 1228 y Fp(\()p Fn( )1239 1207 y Fi(\024)p Fl(h)1288 1228 y Fp(\))-13 1351 y(with)16 b Fm(R)e Fp(=)f(1)e Fm(\000)g(L)16 b Fp(and)h Fm(L)f Fp(is)g(de\014ned)g(as)g(in)g(eq.\(42\)\(43\))g(with) g Fn( )1213 1327 y Fi(\006)p Fl(;)p Fi(\024)p Fj(0)1211 1364 y Fl(k)q Fj(+)p Fl(~)-18 b(!)r(p)1299 1370 y Fe(F)1324 1364 y Fl(;~)g(!)q(;\033)1406 1351 y Fp(replaced)15 b(b)o(y)h Fn( )1700 1327 y Fi(\006)p Fl(;)p Fi(\024)p Fl(h)1698 1364 y(k)q Fj(+)p Fl(~)-18 b(!)q(p)1785 1370 y Fe(F)1810 1364 y Fl(;~)g(!)q(;\033)-13 1429 y Fp(and)18 b Fn(f)112 1411 y Fl(m;)p Fj(0)107 1442 y Fl(n)173 1429 y Fp(\()p Fn(K)233 1436 y Fl(m)266 1429 y Fn(;)8 b Fp(\012)323 1436 y Fl(m)357 1429 y Fp(\))17 b(b)o(y)g Fn(f)491 1411 y Fl(m;h)486 1442 y(n)555 1429 y Fp(\()p Fn(K)615 1436 y Fl(m)648 1429 y Fn(;)8 b Fp(\012)705 1436 y Fl(m)739 1429 y Fp(\).)24 b(Then)17 b Fm(L)p Fn(V)998 1411 y Fl(h)1020 1429 y Fp(\()p Fn( )1073 1411 y Fi(\024)p Fl(h)1122 1429 y Fp(\))h(is)f(giv)o(en)f(b)o(y)h(eq.\(45\))g(with)g Fn(n)1712 1436 y Fj(0)1732 1429 y Fn(;)8 b(a)1780 1436 y Fj(0)1799 1429 y Fn(;)g(z)1844 1436 y Fj(0)1863 1429 y Fn(;)-13 1507 y(i)4 1514 y Fj(0)23 1507 y Fn(;)g(t)63 1514 y Fj(0)82 1507 y Fn(;)g(s)127 1514 y Fj(0)147 1507 y Fn(;)g(\025)197 1514 y Fl(i;)p Fj(0)239 1507 y Fp(,)20 b Fn(i)f Fp(=)g(1)p Fn(;)8 b(:)g(:)g(:)g(;)g Fp(6,)20 b(replaced)f(b)o(y)g Fn(\015)853 1489 y Fl(h)875 1507 y Fn(n)904 1514 y Fl(h)927 1507 y Fn(;)8 b(a)975 1514 y Fl(h)996 1507 y Fn(;)g(z)1041 1514 y Fl(h)1063 1507 y Fn(;)g(i)1102 1514 y Fl(h)1124 1507 y Fn(;)g(t)1164 1514 y Fl(h)1186 1507 y Fn(;)g(\015)1236 1489 y Fl(h)1258 1507 y Fn(s)1281 1514 y Fl(h)1304 1507 y Fn(;)g(\025)1354 1514 y Fl(i;h)1398 1507 y Fp(;)21 b(the)e(reason)h(for)g(whic)o(h)-13 1586 y(w)o(e)13 b(write)h(the)g(terms)e(m)o(ultipling)f Fn(F)665 1568 y Fi(\024)p Fl(h)658 1598 y(\027)729 1586 y Fp(and)j Fn(F)860 1568 y Fi(\024)p Fl(h)853 1598 y(\033)924 1586 y Fp(as)g Fn(\015)1009 1568 y Fl(h)1032 1586 y Fn(s)1055 1593 y Fl(h)1091 1586 y Fp(and)h Fn(\015)1212 1568 y Fl(h)1234 1586 y Fn(n)1263 1593 y Fl(h)1300 1586 y Fp(instead)f(of)g (simply)e Fn(s)1696 1593 y Fl(h)1732 1586 y Fp(and)j Fn(n)1854 1593 y Fl(h)-13 1664 y Fp(will)d(b)q(e)i(discussed)g(after)g (eq.\(52\).)20 b(The)14 b(constan)o(ts)g Fn(n)985 1671 y Fl(h)1008 1664 y Fn(;)8 b(a)1056 1671 y Fl(h)1078 1664 y Fn(;)g(z)1123 1671 y Fl(h)1145 1664 y Fn(;)g(i)1184 1671 y Fl(h)1205 1664 y Fn(;)g(t)1245 1671 y Fl(h)1267 1664 y Fn(;)g(s)1312 1671 y Fl(h)1335 1664 y Fn(\025)1363 1671 y Fl(i;h)1421 1664 y Fp(will)13 b(b)q(e)h(called)e Fg(running)-13 1742 y(c)n(oupling)18 b(c)n(onstants)f Fp(and)g(will)e(b)q(e)i(denoted)f(b)o(y)g Fn(v)920 1749 y Fl(j;h)966 1742 y Fp(.)10 1820 y(It)f(is)g(p)q(ossible)g(to)h(c)o (hec)o(k,)d(see)i(for)g(instance)g(\(B.G.1\),)f(that)h(the)g (e\013ectiv)o(e)f(p)q(oten)o(tial)g Fn(V)1645 1802 y Fl(h)1667 1820 y Fp(\()p Fn( )1720 1802 y Fi(\024)p Fl(h)1769 1820 y Fp(\))h(can)-13 1899 y(b)q(e)h(written)g(in)g(terms)f(of)h Fg(tr)n(e)n(es)g Fp(as)h(follo)o(ws.)10 1977 y(W)l(e)12 b(call)f Fn(\034)196 1984 y Fl(n)232 1977 y Fp(the)g(set)h(of)g(all)f (the)h Fg(lab)n(ele)n(d)i(tr)n(e)n(es)f(with)h Fn(n)g Fg(end)g(p)n(oints)f Fn(\034)19 b Fm(2)c Fn(\034)1318 1984 y Fl(n)1353 1977 y Fp(that)d(can)g(b)q(e)g(costructed)g(as)-13 2055 y(follo)o(ws)f(\(see)f(also)i(the)e(picture\).)19 b(Dra)o(w)11 b(on)h(the)f(\()p Fn(x;)d(y)r Fp(\))i(plane)g(v)o(ertical) g(lines)g(at)h Fn(x)j Fp(=)f Fn(k)r(;)8 b(k)r Fp(+1)p Fn(;)g(:)g(:)g(:)h(;)f Fp(0)p Fn(;)g Fp(1.)-13 2133 y(Let)20 b Fn(r)g Fp(\()p Fg(the)h(r)n(o)n(ot)p Fp(\))e(b)q(e)g(a)h(p)q(oin)o(t) g(on)g(the)f(line)g Fn(x)g Fp(=)h Fn(k)r Fp(.)31 b(Starting)20 b(from)e Fn(r)j Fp(dra)o(w)f(an)g(horizon)o(tal)g(line)-13 2212 y(leading)f(to)i(a)f(p)q(oin)o(t)g Fn(v)421 2219 y Fj(0)460 2212 y Fp(on)h(the)e(line)g Fn(x)h Fp(=)g Fn(k)846 2219 y Fl(v)863 2224 y Ff(0)903 2212 y Fp(=)g Fn(k)15 b Fp(+)f(1.)32 b(Cho)q(ose)22 b Fn(s)1320 2219 y Fl(v)1337 2224 y Ff(0)1376 2212 y Fm(\025)e Fp(0)g(and)h(dra)o(w)f Fn(s)1725 2219 y Fl(v)1742 2224 y Ff(0)1781 2212 y Fp(lines)-13 2290 y(starting)14 b(from)f Fn(v)303 2297 y Fj(0)336 2290 y Fp(leading)g(to)i Fn(s)582 2297 y Fl(v)616 2290 y Fp(p)q(oin)o(ts)f Fn(v)784 2297 y Fj(1)803 2290 y Fn(;)8 b(:)g(:)g(:)g(;)g(v)937 2297 y Fl(s)953 2301 y Fe(v)987 2290 y Fg(i.e.)17 b Fp(the)d(lines)f(cannot)h(go)h(bac)o(k.)20 b(Do)14 b(the)g(same)-13 2368 y(thing)j(starting)i(with)e(the)g(p)q (oin)o(ts)h Fn(v)668 2375 y Fl(i)699 2368 y Fp(and)h(go)f(on)g (recursiv)o(ely)l(.)23 b(A)17 b(p)q(oin)o(t)g Fn(v)i Fp(is)f(called)e(an)i Fg(end)i(p)n(oint)-13 2446 y Fp(if)d Fn(s)56 2453 y Fl(v)94 2446 y Fp(=)f(0,)j Fg(i.e.)f Fp(if)f(there)h(is) f(no)i(line)e(starting)h(from)f(this)h(p)q(oin)o(t.)27 b(Moreo)o(v)o(er)17 b(a)h(p)q(oin)o(t)g Fn(v)i Fp(is)e(a)g Fg(trivial)-13 2525 y(vertex)g Fp(if)e Fn(s)198 2532 y Fl(v)234 2525 y Fp(=)f(1)i(and)g(a)h Fg(non)g(trivial)h(vertex)f Fp(if)e Fn(s)918 2532 y Fl(v)954 2525 y Fm(\025)e Fp(2.)24 b(Finally)15 b(if)i Fn(h)1309 2532 y Fl(v)1344 2525 y Fp(=)e(1)i(then)g Fn(v)h Fp(is)f(necessarely)-13 2603 y(an)j(end)f(p)q(oin)o(t.)31 b(Clearly)18 b(this)h(pro)q(cess)h(ends)g (when)f(all)g(the)g(reac)o(hed)g(p)q(oin)o(ts)g(are)h(end)f(p)q(oin)o (ts.)31 b(A)907 2728 y(17)p eop %%Page: 18 18 18 17 bop 729 251 a Fb(r)513 323 y(r)1233 386 y(r)657 467 y(r)1233 170 y(r)1305 197 y(r)1449 260 y(r)1521 287 y(r)1449 143 y(r)1377 233 y(r)1521 332 y(r)1449 368 y(r)1377 404 y(r)1449 440 y(r)-84 b(r)g(r)1233 251 y(r)1089 296 y(r)1161 341 y(r)1017 242 y(r)1521 557 y(r)1449 539 y(r)1377 521 y(r)1305 503 y(r)1233 485 y(r)1161 467 y(r)1089 449 y(r)1017 431 y(r)945 413 y(r)873 395 y(r)801 377 y(r)729 359 y(r)657 341 y(r)369 323 y(r)204 b(r)-84 b(r)801 179 y(r)873 107 y(r)945 197 y(r)1161 143 y(r)1305 107 y(r)1233 125 y(r)1089 161 y(r)1017 179 y(r)873 215 y(r)801 233 y(r)657 287 y(r)513 323 y(r)g(r)1377 485 y(r)729 251 y Fa(\000)771 210 y(\000)812 168 y(\000)832 149 y(\000)p 1305 442 144 4 v 1305 440 a(\010)1346 419 y(\010)1388 399 y(\010)1430 378 y(\010)1471 357 y(\010)1479 353 y(\010)1161 341 y(\034)1194 300 y(\034)1200 293 y(\034)945 225 y(Q)987 253 y(Q)1028 280 y(Q)1070 308 y(Q)1111 336 y(Q)1153 363 y(Q)1194 391 y(Q)1236 419 y(Q)1277 446 y(Q)1319 474 y(Q)1335 485 y(Q)1377 233 y(\034)1410 192 y(\034)1416 185 y(\034)1161 160 y(a)1203 177 y(a)1244 193 y(a)1286 210 y(a)1327 226 y(a)1369 243 y(a)1410 260 y(a)1452 276 y(a)1479 287 y(a)585 334 y(X)627 344 y(X)668 354 y(X)710 365 y(X)751 375 y(X)793 386 y(X)834 396 y(X)876 406 y(X)917 417 y(X)959 427 y(X)1000 437 y(X)1042 448 y(X)1084 458 y(X)1125 469 y(X)1167 479 y(X)1208 489 y(X)1250 500 y(X)1291 510 y(X)1333 520 y(X)1374 531 y(X)1416 541 y(X)1457 552 y(X)1479 557 y(X)585 365 y(A)606 406 y(A)627 448 y(A)637 467 y(A)729 251 y(\030)771 241 y(\030)812 231 y(\030)854 220 y(\030)895 210 y(\030)937 199 y(\030)978 189 y(\030)1020 179 y(\030)1061 168 y(\030)1103 158 y(\030)1144 148 y(\030)1186 137 y(\030)1227 127 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V 440 309 V 440 323 V 440 338 V 440 352 V 440 366 V 440 381 V 440 395 V 440 410 V 440 424 V 440 438 V 440 453 V 440 467 V 440 482 V 440 496 V 440 510 V 440 525 V 440 539 V 440 554 V 440 568 V 440 582 V 440 597 V 440 611 V 368 7 V 368 21 V 368 35 V 368 50 V 368 64 V 368 79 V 368 93 V 368 107 V 368 122 V 368 136 V 368 151 V 368 165 V 368 179 V 368 194 V 368 208 V 368 223 V 368 237 V 368 251 V 368 266 V 368 280 V 368 294 V 368 309 V 368 323 V 368 338 V 368 352 V 368 366 V 368 381 V 368 395 V 368 410 V 368 424 V 368 438 V 368 453 V 368 467 V 368 482 V 368 496 V 368 510 V 368 525 V 368 539 V 368 554 V 368 568 V 368 582 V 368 597 V 368 611 V 561 638 a Fn(h)589 645 y Fl(v)386 638 y Fn(k)13 b Fp(+)e(1)-141 b Fn(k)1056 b Fp(0)60 b(+1)567 305 y Fn(v)-186 b(v)429 312 y Fj(0)334 332 y Fn(r)-13 812 y Fg(cluster)16 b Fn(v)h Fp(with)e(frequency)g Fn(h)542 819 y Fl(v)578 812 y Fp(is)g(the)g(set)g(of)h(the)f(end-p)q(oin)o(ts)h (reac)o(heable)e(from)h(the)g(v)o(ertex)f Fn(v)j Fp(with)-13 891 y(frequency)f Fn(h)237 898 y Fl(v)258 891 y Fp(;)i(and)g(the)f (tree)g(pro)o(vides)g(an)i(organization)f(of)g(the)g(endp)q(oin)o(ts)g (in)o(to)f(a)h(hierarc)o(h)o(y)e(of)-13 969 y(clusters.)j(Eac)o(h)14 b(non)f(trivial)f(or)i(trivial)d(v)o(ertex)h(b)q(ear)i(a)f(lab)q(el)g Fm(R)g Fp(except)f Fn(v)1364 976 y Fj(0)1397 969 y Fp(\(see)g(the)h (picture\))f(whic)o(h)-13 1047 y(can)h(b)q(ear)g(either)f(a)h(lab)q(el) f Fm(R)h Fp(or)g(a)h(lab)q(el)e Fm(L)p Fp(.)20 b(T)l(o)14 b(eac)o(h)e(tree)g(w)o(e)g(asso)q(ciate)i(a)f(term)e Fn(V)1525 1029 y Fl(k)1546 1047 y Fp(\()p Fn(\034)s(;)d( )1645 1029 y Fi(\024)p Fl(k)1693 1047 y Fp(\))13 b(de\014ned)-13 1125 y(recursiv)o(ely)i(as)i(follo)o(ws.)24 b(If)17 b Fn(\034)23 b Fp(has)18 b(only)f(one)g(end-p)q(oin)o(t)h(with)f (frequency)f Fn(k)e Fp(+)d(1)18 b(then)f Fn(V)1689 1107 y Fl(k)1710 1125 y Fp(\()p Fn(\034)s(;)8 b( )1809 1107 y Fi(\024)p Fl(k)1858 1125 y Fp(\))-13 1204 y(is)18 b(equal)g(to)g(one) h(of)f(the)g(term)f(of)h(eq.\(45\))g(with)e Fn(~)-22 b(v)944 1211 y Fl(k)983 1204 y Fp(instead)18 b(of)f Fn(~)-22 b(v)1235 1211 y Fj(0)1273 1204 y Fp(or,)18 b(only)g(if)g Fn(k)h Fp(=)e(0,)i(one)f(of)h(the)-13 1282 y(monomial)c(in)i Fm(R)p Fn(V)353 1264 y Fj(0)372 1282 y Fp(.)25 b(W)l(e)18 b(attac)o(h)f(a)h(lab)q(el)f(to)h(eac)o(h)f(endp)q(oin)o(t)h(of)f(the)h (tree)e(to)i(distinguish)g(among)-13 1360 y(these)e(p)q(ossibilities.)k (Otherwise)453 1466 y Fn(V)492 1446 y Fl(k)514 1466 y Fp(\()p Fn(\034)s(;)8 b( )613 1446 y Fi(\024)p Fl(k)661 1466 y Fp(\))13 b(=)h Fm(O)808 1433 y Fp(1)p 791 1455 57 2 v 791 1500 a Fn(s)814 1507 y Fl(v)835 1500 y Fp(!)853 1466 y Fm(E)883 1446 y Fl(T)879 1479 y(k)q Fj(+1)946 1466 y Fp([)p Fn(V)999 1446 y Fl(k)q Fj(+1)1065 1466 y Fp(\()p Fn(\034)1111 1446 y Fj(1)1131 1466 y Fn(;)8 b( )1187 1446 y Fi(\024)p Fl(k)q Fj(+1)1280 1466 y Fp(\))p Fn(;)g(:)g(:)g(:)o Fp(\)])379 b(\(50\))-13 1579 y(where)13 b Fm(O)i Fp(is)f Fm(L)h Fp(or)f Fm(R)g Fp(if)g(the)f(v)o(ertex)g Fn(v)i Fp(b)q(ears)g(an)f Fm(L)h Fp(or)f Fm(R)g Fp(lab)q(el,)g Fn(n)g Fm(\025)f Fp(2)i(,)f Fn(\034)1371 1561 y Fj(1)1399 1579 y Fn(:)8 b(:)g(:)f(\034)1491 1561 y Fl(s)1507 1565 y Fe(v)1542 1579 y Fp(are)14 b(the)g(subtrees)-13 1658 y(starting)20 b(from)f Fn(v)i Fp(and)f(the)f(sym)o(b)q(ols)g Fm(E)741 1640 y Fl(T)737 1670 y(h)789 1658 y Fp(denote)h(the)f (truncated)h(exp)q(ectations)f(with)h(resp)q(ect)f(to)-13 1736 y(an)i(in)o(tegration)g(with)g(propagator)i Fn(g)707 1718 y Fl(h)730 1736 y Fp(.)35 b(W)l(e)21 b(ha)o(v)o(e)f(that)i Fm(O)g Fp(is)f(equal)f(to)i Fm(L)f Fp(only)g(if)f Fn(v)k Fp(=)e Fn(v)1758 1743 y Fj(0)1798 1736 y Fp(and)-13 1814 y(the)16 b(tree)f(con)o(tributes)g(to)i(the)f(lo)q(cal)g(part)g(of)g (the)g(e\013ectiv)o(e)e(p)q(oten)o(tial.)21 b(W)l(e)16 b(also)h(asso)q(ciate)g(to)f(eac)o(h)-13 1892 y(\014eld)f(a)h(lab)q (els)f Fn(f)5 b Fp(,)16 b Fn(f)j Fp(=)14 b(1)p Fn(;)8 b(:)g(:)g(:)g(;)g(n)585 1899 y Fl(\034)622 1892 y Fp(where)15 b Fn(n)791 1899 y Fl(\034)828 1892 y Fp(is)h(the)f(n)o(um)o(b)q(er)f (of)i(the)f(\014elds)h(asso)q(ciated)g(with)g(all)f(the)-13 1971 y(endp)q(oin)o(ts)h(of)g(the)f(tree.)21 b(T)l(o)16 b(ev)o(ery)e(\014eld)h(with)h(lab)q(el)f Fn(f)22 b Fp(corresp)q(onds)17 b(a)f(momen)o(tum)11 b Fn(k)r Fp(\()p Fn(f)5 b Fp(\))16 b(and)h(the)-13 2049 y(indices)d Fn(~)-24 b(!)r Fp(\()p Fn(f)5 b Fp(\))p Fn(;)j(\033)r Fp(\()p Fn(f)d Fp(\))p Fn(;)j(")p Fp(\()p Fn(f)d Fp(\))13 b(=)h Fm(\006)p Fp(1)h(and,)g(also,) h(the)e(index)g Fn(s)p Fp(\()p Fn(f)5 b Fp(\))14 b(=)g(0)p Fn(;)8 b Fp(1)p Fn(;)g Fp(2)16 b(allo)o(wing)e(us)i(to)f(distinguish) -13 2127 y(the)h(three)g(p)q(ossibilities)f Fn( )492 2102 y Fl(")p Fj(\()p Fl(f)t Fj(\))490 2142 y Fl(k)q Fj(\()p Fl(f)t Fj(\)+)p Fl(~)-18 b(!)607 2148 y Fe(f)627 2142 y Fl(p)645 2148 y Fe(F)670 2142 y Fl(;~)g(!)q Fj(\()p Fl(f)t Fj(\))p Fl(;\033)q Fj(\()p Fl(f)t Fj(\))832 2127 y Fn(;)8 b(E)s Fp(\()909 2114 y Fn(~)912 2127 y(k)r Fp(\))p Fn( )992 2102 y Fl(")p Fj(\()p Fl(f)t Fj(\))990 2142 y Fl(k)q Fj(\()p Fl(f)t Fj(\))p Fl(~)-18 b(!)1080 2148 y Fe(f)1099 2142 y Fl(p)1117 2148 y Fe(F)1142 2142 y Fl(;~)g(!)q Fj(\()p Fl(f)t Fj(\))p Fl(;\033)q Fj(\()p Fl(f)t Fj(\))1304 2127 y Fn(;)8 b Fm(\000)p Fn(ik)1407 2134 y Fj(0)1427 2127 y Fn( )1461 2102 y Fl(")p Fj(\()p Fl(f)t Fj(\))1459 2142 y Fl(k)q Fj(\()p Fl(f)t Fj(\))p Fl(~)-18 b(!)1549 2148 y Fe(f)1567 2142 y Fl(p)1585 2148 y Fe(F)1611 2142 y Fl(;~)g(!)q Fj(\()p Fl(f)t Fj(\))p Fl(;\033)q Fj(\()p Fl(f)t Fj(\))1773 2127 y Fp(.)22 b(W)l(e)-13 2205 y(call)15 b Fn(I)98 2212 y Fl(v)115 2217 y Ff(0)151 2205 y Fp(the)h(set)g(of)g Fn(f)22 b Fp(lab)q(els.)10 2284 y(It)16 b(is)g(p)q(ossible)h(to)f(c)o(hec)o(k)f(that)h(the)g (e\013ectiv)o(e)f(p)q(oten)o(tial)h(eq\(39\))g(can)h(b)q(e)f(written)g (as)607 2401 y Fn(V)646 2380 y Fl(k)667 2401 y Fp(\()p Fn( )720 2380 y Fi(\024)p Fl(k)769 2401 y Fp(\))d(=)869 2347 y Fi(1)857 2359 y Fk(X)853 2450 y Fl(n)p Fj(=1)938 2359 y Fk(X)928 2447 y Fl(\034)t Fi(2)p Fl(\034)988 2451 y Fe(n)1016 2401 y Fn(V)1056 2380 y Fl(k)1077 2401 y Fp(\()p Fn(\034)s(;)8 b( 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Fp(=)e Fn(v)1159 2429 y Fj(0)1179 2422 y Fp(,)h(while)h(if)f Fn(v)h Fm(\021)e Fn(v)1502 2429 y Fj(0)1538 2422 y Fp(the)i(op)q(eration)h Fm(L)-13 2501 y Fp(or)g Fm(R)h Fp(acts,)f(dep)q(ending)g(on)h(whether)f(it)g (con)o(tributes)f(to)i(the)e(relev)m(an)o(t)h(or)g(to)h(the)e(irrelev)m (an)o(t)g(part)-13 2579 y(of)f(the)g(e\013ectiv)o(e)f(p)q(oten)o(tial.) 907 2728 y(19)p eop %%Page: 20 20 20 19 bop 10 -5 a Fp(W)l(e)23 b(call)f Fg(sc)n(aling)i(dimension)g Fn(D)q Fp(\()p Fn(P)693 2 y Fl(v)710 7 y Ff(0)731 -5 y Fp(\))g(=)h Fm(\000)p Fp(2)16 b(+)969 -38 y Fk(P)1013 6 y Fl(A)p Fi(2)p Fl(P)1085 10 y Fe(v)1101 17 y Ff(0)1123 -5 y Fp(\(1)p Fn(=)p Fp(2)g(+)g Fn(\037)1315 2 y Fl(A)1343 -5 y Fp(\))22 b(where)h Fn(\037)1563 2 y Fl(A)1616 -5 y Fp(=)i(0)e(if)f Fn(A)j Fp(=)-13 74 y Fn( )21 55 y Fl(h)19 86 y(k)q Fj(+)p Fl(~)-18 b(!)q(p)106 92 y Fe(F)132 86 y Fl(;~)g(!;\033)198 74 y Fp(,)19 b Fn(\037)262 81 y Fl(A)308 74 y Fp(=)f(1)i(if)e Fn(A)g Fp(=)g Fn(E)s Fp(\()p Fn(k)r Fp(\))p Fn( )704 55 y 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259 y Fj(\))900 279 y Fn(d)925 286 y Fl(h)948 279 y Fp(\()p Fn(k)992 286 y Fj(1)1012 279 y Fp(\))8 b Fm(\001)g(\001)g(\001)h Fn(d)1131 286 y Fl(h)1153 279 y Fp(\()p Fn(k)1197 287 y Fi(j)p Fl(P)1229 291 y Fe(v)1248 287 y Fi(j)1260 279 y Fp(\))p Fn(f)1308 259 y Fl(P)1330 263 y Fe(v)1350 259 y Fl(;h)1380 263 y Fe(v)1303 292 y Fl(n)1400 279 y Fp(\()p Fn(K)1464 259 y Fl(P)1486 263 y Fe(v)1507 279 y Fn(;)f Fp(\012)1564 259 y Fl(P)1586 263 y Fe(v)1607 279 y Fp(\))164 b(\(52\))-13 442 y(where)16 b Fn(d)153 449 y Fl(h)175 442 y Fp(\()p Fn(k)r Fp(\))h(is)f(the)g(c)o(haracteristic)e(function)i(of)h(the)f (supp)q(ort)h(of)g Fn(h)p Fp(\()p Fn(\015)1331 424 y Fi(\000)p Fj(2)p Fl(h)p Fj(+2)1444 442 y Fp(\()p Fn(k)1490 424 y Fj(2)1488 454 y(0)1520 442 y Fp(+)11 b Fn(E)s Fp(\()1624 429 y Fn(~)1627 442 y(k)s Fp(\))1674 424 y Fj(2)1693 442 y Fp(\)\).)10 520 y(In)21 b(order)g(to)g(motiv)m(ate)e(our)i (de\014nition)f(of)h(lo)q(calization)f(supp)q(ose)i(for)f(a)g(momen)o (t)d(that)j Fm(R)h Fp(=)f Fn(I)-13 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Fl(A)p Fi(2)p Fl(P)1698 1271 y Fe(v)1728 1257 y Fn(\037)1759 1264 y Fl(A)1800 1257 y Fp(=)h(0)-13 1335 y(or)i Fm(j)p Fn(P)91 1342 y Fl(v)112 1335 y Fm(j)e Fp(=)f(2)k(and)326 1302 y Fk(P)370 1345 y Fl(A)p Fi(2)p Fl(P)442 1349 y Fe(v)471 1335 y Fn(\037)502 1342 y Fl(A)544 1335 y Fp(=)d(1.)21 b(Note)16 b(that)g Fn(D)q Fp(\()p Fn(P)969 1342 y Fl(v)991 1335 y Fp(\))g(dep)q(ends)g (only)g(on)g(the)g(n)o(um)o(b)q(er)e(of)i(external)-13 1413 y(with)j Fn(\037)h Fp(=)f(0)h(or)h(1)f(line)e Fn( )490 1395 y Fi(\024)p Fl(h)537 1399 y Fe(v)558 1413 y Fp(\()p Fn(P)608 1420 y Fl(v)629 1413 y Fp(\):)28 b(this)20 b(is)f(due)h(to)g (the)f(fact)h(that)g(w)o(e)f(write)g(the)h(co)q(e\016cien)o(ts)e(of)-13 1492 y Fn(F)26 1473 y Fi(\024)p Fl(h)19 1504 y(\027)75 1492 y Fn(;)8 b(F)136 1473 y Fi(\024)p Fl(h)129 1504 y(\033)201 1492 y Fp(as)17 b Fn(\015)289 1473 y Fl(h)312 1492 y Fn(n)341 1499 y Fl(h)379 1492 y Fp(and)g Fn(\015)502 1473 y Fl(h)525 1492 y Fn(\033)553 1499 y Fl(h)575 1492 y Fp(.)10 1570 y(Lik)o(e)h(in)g(\(B.G.\))f(one)h(could)g(de\014ne)g(as) h("relev)m(an)o(t)e(part")i(of)g(the)f(e\013ectiv)o(e)e(p)q(oten)o (tial)i(the)g(sum)f(of)-13 1648 y(its)f(quadratic)f(and)i(quartic)e (parts)i(in)f(the)f(\014elds.)21 b(Ho)o(w)o(ev)o(er)14 b(suc)o(h)i(de\014nitions)g(w)o(ould)g(still)f(con)o(tain)-13 1726 y(irrelev)m(an)o(t)j(terms.)28 b(This)20 b(can)f(b)q(e)g(easily)g (understo)q(o)q(d)i(b)o(y)d(remarking)g(that)i(for)f Fn(h)g Fp(suitable)g(small)-13 1805 y(the)d(con)o(tributions)g(to)g (the)g(e\013ectiv)o(e)f(p)q(oten)o(tial)h Fn(V)945 1786 y Fl(h)984 1805 y Fp(ha)o(ving)g(forms:)225 1889 y Fk(Z)294 1893 y Fj(4)277 1906 y Fk(Y)275 1997 y Fl(i)p Fj(=1)340 1947 y Fn(dk)390 1954 y Fl(i)405 1947 y Fn(f)434 1927 y Fj(4)p Fl(;h)429 1959 y(n)484 1947 y Fp(\()p Fn(K)544 1954 y Fj(4)564 1947 y Fp(;)8 b(\012)621 1954 y Fj(4)641 1947 y Fp(\))p Fn(\016)r Fp(\()p Fn(k)728 1954 y Fj(1)758 1947 y Fp(+)j Fn(k)832 1954 y Fj(2)864 1947 y Fm(\000)f Fn(k)938 1954 y Fj(3)970 1947 y Fm(\000)g Fn(k)1044 1954 y Fj(4)1076 1947 y Fp(+)h(\()p Fn(~)-24 b(!)1174 1954 y Fj(1)1205 1947 y Fp(+)11 b Fn(~)-24 b(!)1284 1954 y Fj(2)1315 1947 y Fm(\000)11 b Fn(~)-24 b(!)1395 1954 y Fj(3)1426 1947 y Fm(\000)11 b Fn(~)-24 b(!)1506 1954 y Fj(4)1526 1947 y Fp(\))p Fn(p)1569 1954 y Fl(F)1610 1947 y Fp(+)1664 1913 y(2)p Fn(n\031)p 1664 1935 83 2 v 1692 1981 a(a)1751 1947 y Fp(\))225 2064 y Fn( )259 2041 y Fj(+)p Fl(;)p Fi(\024)p Fl(h)257 2077 y(k)275 2082 y Ff(1)292 2077 y Fj(+)p Fl(~)-18 b(!)341 2082 y Ff(1)359 2077 y Fl(p)377 2083 y Fe(F)402 2077 y Fl(;~)g(!)434 2082 y Ff(1)451 2077 y Fl(;\033)484 2064 y Fn( )518 2041 y Fj(+)p Fl(;)p Fi(\024)p Fl(h)516 2077 y(k)534 2082 y Ff(2)552 2077 y Fj(+)p Fl(~)g(!)601 2082 y Ff(2)618 2077 y Fl(p)636 2083 y Fe(F)661 2077 y Fl(;~)g(!)693 2082 y Ff(2)710 2077 y Fl(;\033)741 2068 y Fh(0)755 2064 y Fn( )789 2041 y Fi(\000)p Fl(;)p Fi(\024)p Fl(h)787 2077 y(k)805 2082 y Ff(3)822 2077 y Fj(+)p Fl(~)g(!)871 2082 y Ff(3)889 2077 y Fl(p)907 2083 y Fe(F)932 2077 y Fl(;~)g(!)964 2082 y Ff(3)981 2077 y Fl(;\033)1014 2064 y Fn( )1048 2041 y Fi(\000)p Fl(;)p Fi(\024)p Fl(h)1046 2077 y(k)1064 2082 y Ff(4)1082 2077 y Fj(+)p Fl(~)g(!)1131 2082 y Ff(4)1148 2077 y Fl(p)1166 2083 y Fe(F)1192 2077 y Fl(;~)g(!)1224 2082 y Ff(4)1241 2077 y Fl(;\033)-13 2192 y Fp(or)66 2260 y Fk(Z)116 2319 y Fn(dk)166 2326 y Fj(1)186 2319 y Fn(dk)236 2326 y Fj(2)257 2319 y Fn(\016)r Fp(\()p Fn(k)325 2326 y Fj(1)355 2319 y Fm(\000)11 b Fn(k)430 2326 y Fj(2)461 2319 y Fp(+)g(\()p Fn(~)-24 b(!)559 2326 y Fj(1)590 2319 y Fm(\000)11 b Fn(~)-24 b(!)670 2326 y Fj(2)690 2319 y Fp(\))p Fn(p)733 2326 y Fl(F)774 2319 y Fp(+)11 b(2)p Fn(n\031)r(=a)p Fp(\))p Fn(f)1004 2298 y Fj(2)1024 2319 y Fp(\()p Fn(K)1084 2326 y Fj(2)1104 2319 y Fp(;)d(\012)1161 2326 y Fj(2)1181 2319 y Fp(\))p Fn( )1234 2295 y Fj(+)p Fl(;)p Fi(\024)p Fl(h)1232 2332 y(k)1250 2337 y Ff(1)1267 2332 y Fj(+)p Fl(~)-18 b(!)1316 2337 y Ff(1)1334 2332 y Fl(p)1352 2338 y Fe(F)1377 2332 y Fl(;~)g(!)1409 2337 y Ff(1)1426 2332 y Fl(;\033)1460 2319 y Fn( )1494 2295 y Fi(\000)p Fl(;)p Fi(\024)p Fl(h)1492 2332 y(k)1510 2337 y Ff(2)1527 2332 y Fj(+)p Fl(~)g(!)1576 2337 y Ff(2)1594 2332 y Fl(p)1612 2338 y Fe(F)1637 2332 y Fl(;~)g(!)1669 2337 y Ff(2)1686 2332 y Fl(;\033)1790 2319 y Fp(\(53\))-13 2446 y(are)16 b(v)m(anishing)h(unless)f(\()p Fn(~)-24 b(!)477 2453 y Fj(1)508 2446 y Fp(+)10 b Fn(~)-24 b(!)586 2453 y Fj(2)617 2446 y Fm(\000)11 b Fn(~)-24 b(!)697 2453 y Fj(3)728 2446 y Fm(\000)10 b Fn(~)-24 b(!)807 2453 y Fj(4)827 2446 y Fp(\))p Fn(p)870 2453 y Fl(F)911 2446 y Fp(+)965 2427 y Fj(2)p Fl(n\031)p 965 2435 61 2 v 986 2464 a(a)1044 2446 y Fp(=)14 b(0)i(in)g(the)g(\014rst)h(case)f(and)g(\()p Fn(~)-24 b(!)1624 2453 y Fj(1)1655 2446 y Fm(\000)11 b Fn(~)-24 b(!)1735 2453 y Fj(2)1755 2446 y Fp(\))p Fn(p)1798 2453 y Fl(F)1839 2446 y Fp(+)-13 2525 y(2)p Fn(n\031)r(=a)14 b Fp(=)f(0)k(in)f(the)g(second)g(as)h(the)f(delta's)g(in)g(the)g(ab)q (o)o(v)o(e)g(equations)h(cannot)f(b)q(e)h(satis\014ed)f(for)h(the)-13 2603 y(supp)q(ort)g(prop)q(erties)g(of)f(the)g(\014els.)907 2728 y(20)p eop %%Page: 21 21 21 20 bop 10 -5 a Fp(It)17 b(is)h(p)q(ossible)f(to)h(c)o(hec)o(k)e(b)o (y)h(a)h(standard)h(calculation)d(that,)i(with)g(the)f(de\014nition)g (of)h Fm(L)p Fn(;)8 b Fm(R)17 b Fp(giv)o(en)-13 74 y(b)o(y)c (eq.\(42\)\(43\))g(the)g(size)g(of)g(the)g(generic)f(F)l(eynman)g (graph)i(con)o(tribuing)f(to)h(the)f(e\013ectiv)o(e)e(p)q(oten)o(tial) -13 152 y(de\014ned)16 b(ab)q(o)o(v)o(e)g(is)g(b)q(ounded)h(b)o(y:)517 275 y Fm(jj)p Fn(f)574 255 y Fl(P)596 259 y Fe(v)615 255 y Fl(;h)645 259 y Fe(v)569 287 y Fl(n)666 275 y Fm(jj)c Fn(<)h(C)798 255 y Fl(m)831 275 y Fn(")854 255 y Fl(m)895 234 y Fk(Y)913 321 y Fl(v)957 275 y Fn(\015)985 255 y Fi(\000)p Fj(\()p Fl(h)1046 259 y Fe(v)1065 255 y Fi(\000)p Fl(h)1112 264 y Fe(v)1129 257 y Fh(0)1142 255 y Fj(\)\()p Fl(D)q Fj(\()p Fl(P)1236 259 y Fe(v)1254 255 y Fj(\)+)p Fl(z)1311 259 y Fe(v)1330 255 y Fj(\))1790 275 y Fp(\(54\))-13 412 y(where)23 b Fn(D)q Fp(\()p Fn(P)226 419 y Fl(v)248 412 y Fp(\))16 b(+)g Fn(z)360 419 y Fl(v)406 412 y Fn(>)27 b Fp(0)d(\(the)f Fm(R)h Fp(op)q(eration)g(w)o(as)g(de\014ned)g(in)f (order)h(to)g(mak)o(e)d(true)j(suc)o(h)f(an)-13 490 y(inequalit)o(y\).) c(By)c(rep)q(eating)i(the)f(estimates)e(in)i(\(B.G.P)l(.S.\)\(B.M.\))d (it)j(is)g(easy)g(to)g(see)g(that)h(eq.\(54\))-13 568 y(implies)g(that)j Fm(j)p Fn(V)318 550 y Fj(\()p Fl(k)q Fj(\))367 568 y Fp(\()p Fn(\034)s(;)8 b( )466 550 y Fi(\024)p Fl(k)514 568 y Fp(\))p Fm(j)20 b(\024)f Fn(")648 550 y Fl(n)672 568 y Fn(c)693 550 y Fl(n)736 568 y Fp(so)h(that)h(if)e Fn(")h Fp(is)f(small)f(enough)j(the)e(partition)h(function)g(is)-13 646 y(analytic)15 b(in)h(its)g(argumen)o(t.)10 725 y(The)k(problem)e (is)h(that)h Fg(we)h(do)g(not)g(exp)n(e)n(ct)g(that)f Fm(j)p Fn(v)989 732 y Fl(h;i)1032 725 y Fm(j)g(\024)f Fn(")p Fp(,)h(and)g(this)f(b)q(ecause)h(a)g(second)f(order)-13 803 y(computation)c(\(in)h(the)g(spinless)g(case)g(for)h(semplicit)n (y\))c(sho)o(ws)18 b(that)e Fn(\025)1309 810 y Fl(h)p Fi(\000)p Fj(1)1391 803 y Fp(=)e Fn(\025)1471 810 y Fl(h)1510 803 y Fp(and)46 919 y(1.)25 b Fn(z)132 926 y Fl(h)p Fi(\000)p Fj(1)213 919 y Fp(=)14 b Fn(z)288 926 y Fl(h)321 919 y Fp(+)d Fn(\014)398 926 y Fj(1)417 919 y Fn(\025)445 900 y Fj(2)445 931 y Fl(h)468 919 y Fp(;)105 b Fn(a)613 926 y Fl(h)p Fi(\000)p Fj(1)694 919 y Fp(=)14 b Fn(a)772 926 y Fl(h)805 919 y Fp(+)d Fn(\014)882 926 y Fj(1)901 919 y Fn(\025)929 900 y Fj(2)929 931 y Fl(h)46 1037 y Fp(2.)25 b Fn(s)132 1044 y Fl(h)p Fi(\000)p Fj(1)213 1037 y Fp(=)14 b Fn(\015)s(s)316 1044 y Fl(h)349 1037 y Fp(+)d Fn(\015)s(s)449 1044 y Fl(h)472 1037 y Fn(\014)500 1044 y Fj(3)519 1037 y Fn(\025)547 1044 y Fl(h)-13 1153 y Fp(with)k Fn(\014)125 1160 y Fj(3)144 1153 y Fn(;)8 b(\014)194 1160 y Fj(1)227 1153 y Fn(>)14 b Fp(0,)i(so)g(that,)g(ev)o (en)e(if)h(a)h(third)f(order)h(computation)f(w)o(ould)g(sho)o(w)h(that) g Fn(\025)1631 1160 y Fl(h)1668 1153 y Fm(!)1718 1160 y Fl(h)p Fi(!\0001)1852 1153 y Fp(0)-13 1231 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y(an)e(indication)e(that)i(the)f(Sc)o(h)o(winger)g(function)g(b)q(eha)o (viour)g(for)h(large)f(distances)g(is)g(di\013eren)o(t)g(in)g(the)-13 1622 y(in)o(teracting)h(case)i(resp)q(ect)f(to)h(the)f(free)g(one.)26 b(The)17 b(idea)h(is)f(then)g(to)h(mo)q(dify)f(the)g(de\014nition)g(of) h Fn(V)1854 1604 y Fl(h)-13 1700 y Fp(at)h(eac)o(h)f(step)g(of)h(the)g (renormalization)e(group)i(generalizing)f(the)g(anomalous)h(scaling)g (pro)q(cedure)-13 1779 y(of)d(\(B.G.\))f(in)h(order)h(to)f(tak)o(e)g (in)o(to)g(accoun)o(t)g(the)g(presence)f(of)i Fn(\033)1188 1786 y Fl(h)1210 1779 y Fp(.)-13 1945 y Fc(2.2)69 b(Anomalous)21 b(scaling)-13 2055 y Fp(In)15 b(the)h(preceding)f(section)g(w)o(e)g(sa) o(w)h(that)g(it)g(is)f(p)q(ossible)h(to)g(express)f(the)h(partition)f (function)h(as)g(an)-13 2133 y(analytic)h(function)h(of)h(the)e (running)i(coupling)f(constan)o(ts,)g(if)g(suc)o(h)g(constan)o(ts)h (are)f(small)e(enough.)-13 2212 y(Ho)o(w)o(ev)o(er)k(a)j(second)f (order)g(computation)g(suggests)h(that)g 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b(is)h(p)q(ossible)g(to)g(extend)f(the)907 2728 y(21)p eop %%Page: 22 22 22 21 bop -13 -5 a Fp(metho)q(ds)12 b(so)i(far)f(follo)o(w)o(ed)f(to)h (a)g(more)e(general)i(approac)o(h)h(whic)o(h)e(can)h(tak)o(e)f(in)o(to) g(accoun)o(t)h(a)g(p)q(ossible)-13 74 y(Sc)o(h)o(winger)21 b(function)h(b)q(eha)o(viour)g(mo)q(di\014cation)f(resp)q(ect)h(to)g (the)g(free)f(case.)38 b(Suc)o(h)22 b(approac)o(h)h(is)-13 152 y(called)17 b(\\anomalous)j(scaling")f(and)g(it)f(w)o(as)i(used)e (for)h(the)g(\014rst)g(time)d(in)j(\(W.F.\))e(for)i(the)g(infrared)-13 230 y(problem)12 b(in)i(the)g Fn(\036)340 212 y Fj(4)340 242 y(3)374 230 y Fp(mo)q(del,)f(in)h(\(B.G.\),\(B.G.M.)o(\),\(B.)o (G.P)l(.S\))d(for)j(the)g(study)h(of)f(a)h(system)e(of)h Fn(d)g Fp(=)g(1)-13 308 y(F)l(ermi)g(system)g(and)j(in)f(\(B.G.1\))g (for)g(the)g(Bose)g(condensation)h(problem.)10 387 y(W)l(e)f(write,)g (calling)f Fn(P)7 b Fp(\()p Fn(d )503 368 y Fi(\024)p Fj(0)550 387 y Fp(\))14 b(=)g Fn(P)666 394 y Fl(Z)690 399 y Ff(0)710 387 y Fp(\()p Fn(d )788 368 y Fi(\024)p Fj(0)835 387 y Fp(\))i(with)g Fn(Z)1014 394 y Fj(0)1048 387 y Fp(=)e(1:)203 498 y Fm(N)21 b Fp(=)316 439 y Fk(Z)366 498 y Fn(P)397 505 y Fl(Z)421 510 y Ff(0)441 498 y Fp(\()p Fn(d )519 477 y Fi(\024)p Fj(0)566 498 y Fp(\))p Fn(e)608 477 y Fi(\000)p Fl(V)663 465 y Ff(0)681 477 y Fj(\()695 451 y Fi(p)p 724 451 42 2 v 26 x Fl(Z)748 482 y Ff(0)765 477 y Fl( )789 465 y Fh(\024)p Ff(0)831 477 y Fj(\))860 498 y Fp(=)912 439 y Fk(Z)977 456 y(Y)960 549 y Fl(~)962 558 y(k)q(;~)-18 b(!)q(;\033)1053 498 y Fn(d )1112 474 y Fj(+)p Fl(;)p Fi(\024)p Fj(0)1110 511 y Fl(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)r(;\033)1268 498 y Fn(d )1327 474 y Fi(\000)p Fl(;)p Fi(\024)p Fj(0)1325 511 y Fl(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)1482 498 y Fn(e)1505 477 y Fi(\000)p Fl(V)1561 465 y Ff(0)1578 477 y Fj(\()1592 451 y Fi(p)p 1621 451 V 26 x Fl(Z)1645 482 y Ff(0)1662 477 y Fl( )1686 465 y Fh(\024)p Ff(0)1728 477 y Fj(\))203 672 y Fp(exp)285 585 y Fk(8)285 622 y(<)285 697 y(:)322 672 y Fm(\000)369 631 y Fk(X)373 724 y Fl(\033)o(;~)g(!)438 614 y Fk(Z)487 672 y Fn(dk)r(Z)572 679 y Fj(0)593 672 y Fn(C)628 679 y Fj(0)648 672 y Fp(\()p Fn(k)r Fp(\))718 638 y Fm(\000)p Fn(ik)799 645 y Fj(0)829 638 y Fp(+)876 625 y Fn(~)878 638 y(k)905 620 y Fj(2)936 638 y Fp(+)11 b(2)p Fn(~)-24 b(!)r(\031)1068 625 y(~)1071 638 y(k)p 718 661 381 2 v 787 714 a(\037)p Fp(\()p Fn(~)g(!)866 701 y(~)869 714 y(k)13 b Fp(+)e Fn(p)980 721 y Fl(F)1010 714 y Fp(\))1103 672 y Fn( )1137 649 y Fj(+)p Fl(;)p Fi(\024)p Fj(0)1135 685 y Fl(k)q Fj(+)p Fl(~)-18 b(!)q(\031)q(;~)g(!)r (;\033)1292 672 y Fn( )1326 649 y Fi(\000)p Fl(;)p Fi(\024)p Fj(0)1324 685 y Fl(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)1481 585 y Fk(9)1481 622 y(=)1481 697 y(;)-13 806 y Fp(where)16 b(the)g(grassman)g(in)o(tegration)g(is)g(written)g(formally)e(as)j(an)g (in)o(tegral)f(o)o(v)o(er)-13 884 y Fn(d )46 861 y Fj(+)p Fl(;)p Fi(\024)p Fj(0)44 897 y Fl(k)q Fj(+)p Fl(~)-18 b(!)r(\031)q(;~)g(!)q(;\033)201 884 y Fn(d )260 861 y Fi(\000)p Fl(;)p Fi(\024)p Fj(0)258 897 y Fl(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)r(;\033)434 884 y Fp(times)17 b(the)h(exp)q(onen)o(tial)g(of)h(the)f(appropriate)h(quadratic)f(form.) 27 b(The)19 b(idea)-13 962 y(is)j(no)o(w)g(to)g(extract)f(from)g(the)h (e\013ectiv)o(e)e(p)q(oten)o(tial)h(the)h(terms)e(whic)o(h)i(are)g (di\016cult)e(to)i(con)o(trol,)-13 1041 y Fg(i.e.)17 b Fn(s)94 1048 y Fj(0)114 1041 y Fn(F)153 1023 y Fi(\024)p Fj(0)146 1053 y Fl(\033)222 1041 y Fp(and)23 b Fn(z)346 1048 y Fj(0)365 1041 y Fn(F)404 1018 y Fi(\024)p Fj(0)397 1053 y Fl(\030)473 1041 y Fp(and)g(to)g(put)f(them)f(in)h(the)g (grassmanian)h(in)o(tegration.)39 b(This)23 b(is)f(made)-13 1119 y(writing:)371 1197 y Fm(N)f Fp(=)485 1139 y Fk(Z)534 1197 y Fn(P)565 1204 y Fl(Z)589 1209 y Ff(0)609 1197 y Fp(\()p Fn(d )687 1177 y Fi(\024)p Fj(0)734 1197 y Fp(\))p Fn(e)776 1177 y Fi(\000)p Fl(V)831 1165 y Ff(0)849 1177 y Fj(\()p Fl( )887 1165 y Fh(\024)p Ff(0)928 1177 y Fj(\))958 1197 y Fp(=)1009 1139 y Fk(Z)1059 1197 y Fn(P)1090 1204 y Fl(Z)1114 1209 y Fh(\000)p Ff(1)1158 1197 y Fp(\()p Fn(d )1236 1177 y Fi(\024)p Fj(0)1283 1197 y Fp(\))p Fn(e)1325 1177 y Fi(\000)1357 1168 y Fj(~)1352 1177 y Fl(V)1380 1165 y Ff(0)1398 1177 y Fj(\()p Fl( )1436 1165 y Fh(\024)p Ff(0)1477 1177 y Fj(\))-13 1295 y Fp(where:)254 1406 y Fn(P)285 1413 y Fl(Z)309 1418 y Fh(\000)p Ff(1)352 1406 y Fp(\()p Fn(d )430 1385 y Fi(\024)p Fj(0)477 1406 y Fp(\))14 b(=)577 1365 y Fk(Y)560 1458 y Fl(~)562 1466 y(k)q(;~)-18 b(!)q(;\033)654 1406 y Fn(d )713 1385 y Fj(+)711 1419 y Fl(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)868 1406 y Fn(d )927 1385 y Fi(\000)925 1419 y Fl(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)254 1576 y Fp(exp)328 1515 y Fk(\032)359 1576 y Fm(\000)406 1534 y Fk(X)410 1627 y Fl(\033)o(;~)g(!)475 1517 y Fk(Z)524 1576 y Fn(dk)r(C)611 1583 y Fj(0)631 1576 y Fp(\()p Fn(k)r Fp(\))p Fn(Z)729 1583 y Fi(\000)p Fj(1)777 1576 y Fp(\()p Fn(k)r Fp(\))842 1515 y Fk(\024)869 1542 y Fm(\000)p Fn(ik)950 1549 y Fj(0)980 1542 y Fp(+)1027 1529 y Fn(~)1029 1542 y(k)1056 1524 y Fj(2)1087 1542 y Fp(+)11 b(2)p Fn(~)-24 b(!)s(\031)1220 1529 y(~)1223 1542 y(k)p 869 1564 V 938 1618 a(\037)p Fp(\()p Fn(~)g(!)1017 1605 y(~)1020 1618 y(k)13 b Fp(+)e Fn(p)1131 1625 y Fl(F)1161 1618 y Fp(\))1254 1576 y Fn( )1288 1555 y Fj(+)1286 1589 y Fl(k)q Fj(+)p Fl(~)-18 b(!)r(\031)q(;~)g(!)q (;\033)1443 1576 y Fn( )1477 1555 y Fi(\000)1475 1589 y Fl(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)1644 1576 y Fp(+)254 1717 y Fn(\033)282 1724 y Fi(\000)p Fj(1)329 1717 y Fp(\()p Fn(k)r Fp(\))p Fn( )428 1696 y Fj(+)426 1730 y Fl(k)q Fj(+)p Fl(~)g(!)q(\031)q(;~)g(!)r(;\033)583 1717 y Fn( )617 1696 y Fi(\000)615 1730 y Fl(k)q Fi(\000)p Fl(~)g(!)q(\031)q(;)p Fi(\000)p Fl(~)g(!)r(;\033)799 1656 y Fk(\025\033)1790 1717 y Fp(\(55\))-13 1837 y(with)14 b Fn(Z)129 1844 y Fi(\000)p Fj(1)176 1837 y Fp(\()p Fn(k)r Fp(\))g(=)g Fn(Z)340 1844 y Fj(0)367 1837 y Fp(+)7 b Fn(C)451 1816 y Fi(\000)p Fj(1)447 1848 y(0)498 1837 y Fp(\()p Fn(k)r Fp(\))p Fn(z)586 1844 y Fj(0)605 1837 y Fp(,)14 b Fn(Z)666 1844 y Fi(\000)p Fj(1)714 1837 y Fp(\()p Fn(k)r Fp(\))p Fn(\033)807 1844 y Fi(\000)p Fj(1)854 1837 y Fp(\()p Fn(k)r Fp(\))f(=)h Fn(C)1023 1816 y Fi(\000)p Fj(1)1019 1848 y(0)1070 1837 y Fp(\()p Fn(k)r Fp(\))p Fn(\033)1163 1844 y Fj(0)1196 1837 y Fp(and)1297 1824 y(~)1289 1837 y Fn(V)1329 1819 y Fj(0)1362 1837 y Fp(=)g Fm(L)1455 1824 y Fp(~)1448 1837 y Fn(V)1488 1819 y Fj(0)1514 1837 y Fp(+)7 b(\(1)g Fm(\000)g(L)p Fp(\))p Fn(V)1748 1819 y Fj(0)1782 1837 y Fp(with)235 1966 y Fm(L)276 1953 y Fp(~)269 1966 y Fn(V)309 1945 y Fj(0)329 1966 y Fp(\()p Fn( )382 1945 y Fi(\024)p Fj(0)428 1966 y Fp(\))14 b(=)g Fn(n)542 1973 y Fj(0)562 1966 y Fn(F)601 1945 y Fi(\024)p Fj(0)594 1978 y Fl(\027)658 1966 y Fp(+)d(\()p Fn(a)752 1973 y Fj(0)783 1966 y Fm(\000)f Fn(z)855 1973 y Fj(0)875 1966 y Fp(\))p Fn(F)933 1945 y Fi(\024)p Fj(0)926 1978 y Fl(\013)990 1966 y Fp(+)h Fn(t)1057 1973 y Fj(0)1077 1966 y Fn(F)1116 1945 y Fi(\024)p Fj(0)1109 1978 y Fl(t)1173 1966 y Fp(+)g Fn(i)1239 1973 y Fj(0)1259 1966 y Fn(F)1298 1943 y Fi(\024)p Fj(0)1291 1978 y Fl(\022)1355 1966 y Fp(+)1425 1912 y Fj(6)1404 1924 y Fk(X)1406 2015 y Fl(i)p Fj(=1)1473 1966 y Fn(\025)1501 1973 y Fl(i;)p Fj(0)1543 1966 y Fn(F)1582 1943 y Fi(\024)p Fj(0)1575 1977 y Fl(i)1790 1966 y Fp(\(56\))-13 2096 y(The)16 b(Grassman)h(in)o(tegral)f Fn(P)525 2103 y Fl(Z)549 2108 y Fh(\000)p Ff(1)592 2096 y Fp(\()p Fn(d )670 2078 y Fi(\024)p Fj(0)717 2096 y Fp(\))h(can)f(also)h(b)q(e)g(thougth)g(as)g(an)g(in)o(tegral)f(o)o(v)o (er)f(t)o(w)o(o)i(indep)q(en-)-13 2175 y(den)o(t)f(\014elds)f(adding)i (up)g(to)f Fn( )545 2157 y Fi(\024)p Fj(0)606 2175 y Fp(=)e Fn( )692 2157 y Fi(\024\000)p Fj(1)777 2175 y Fp(+)d Fn( )860 2157 y Fj(0)895 2175 y Fg(i.e.)535 2286 y Fn(P)566 2293 y Fl(Z)590 2298 y Fh(\000)p Ff(1)634 2286 y Fp(\()p Fn(d )712 2265 y Fi(\024)p Fj(0)759 2286 y Fp(\))i(=)h Fn(P)874 2293 y Fl(Z)898 2298 y Fh(\000)p Ff(1)942 2286 y Fp(\()p Fn(d )1020 2265 y Fi(\024\000)p Fj(1)1094 2286 y Fp(\))p Fn(P)1144 2293 y Fl(Z)1168 2298 y Fh(\000)p Ff(1)1212 2286 y Fp(\()p Fn(d )1290 2265 y Fj(0)1309 2286 y Fp(\))-13 2397 y(where)19 b Fn(P)162 2404 y Fl(Z)186 2409 y Fh(\000)p Ff(1)229 2397 y Fp(\()p Fn(d )307 2379 y Fi(\024\000)p Fj(1)382 2397 y Fp(\),)g Fn(P)465 2404 y Fl(Z)489 2409 y Fh(\000)p Ff(1)533 2397 y Fp(\()p Fn(d )611 2379 y Fj(0)631 2397 y Fp(\))g(are)g(equal)g(to)h Fn(P)979 2404 y Fl(Z)1003 2409 y Fh(\000)p Ff(1)1046 2397 y Fp(\()p Fn(d )1124 2379 y Fi(\024)p Fj(0)1171 2397 y Fp(\))g(eq.\(55\))f(with)g Fn(C)1526 2404 y Fj(0)1545 2397 y Fp(\()p Fn(k)r Fp(\))h(replaced)e(b)o(y)-13 2475 y Fn(C)22 2482 y Fi(\000)p Fj(1)69 2475 y Fp(\()p Fn(k)r Fp(\))e(and)h Fn(f)269 2482 y Fj(0)289 2475 y Fp(\()p Fn(k)333 2482 y Fj(0)353 2475 y Fn(;)373 2462 y(~)375 2475 y(k)r Fp(\))f(resp)q(ectiv)o(ely)l(.)j(W)l(e)d(can)g(write)g (then:)429 2586 y Fm(N)21 b Fp(=)543 2527 y Fk(Z)593 2586 y Fn(P)624 2593 y Fl(Z)648 2598 y Fh(\000)p Ff(1)692 2586 y Fp(\()p Fn(d )770 2565 y Fi(\024\000)p Fj(1)844 2586 y Fp(\))871 2527 y Fk(Z)921 2586 y Fn(P)952 2593 y Fl(Z)976 2598 y Fh(\000)p Ff(1)1020 2586 y Fp(\()p Fn(d )1098 2565 y Fj(0)1117 2586 y Fp(\))p Fn(e)1159 2565 y Fi(\000)1191 2557 y Fj(^)1186 2565 y Fl(V)1215 2553 y Ff(0)1232 2565 y Fj(\()1246 2533 y Fm(p)p 1287 2533 66 2 v 1287 2565 a Fl(Z)1311 2570 y Fh(\000)p Ff(1)1353 2565 y Fl( )1377 2553 y Fh(\024)p Ff(0)1418 2565 y Fj(\))1790 2586 y Fp(\(57\))907 2728 y(22)p eop %%Page: 23 23 23 22 bop -13 -5 a Fp(where:)308 79 y Fm(L)349 66 y Fp(^)342 79 y Fn(V)382 58 y Fj(0)402 79 y Fp(\()p Fn( )455 58 y Fi(\024)p Fj(0)501 79 y Fp(\))14 b(=)g Fn(\027)610 86 y Fj(0)630 79 y Fn(F)669 58 y Fi(\024)p Fj(0)662 91 y Fl(\027)726 79 y Fp(+)d Fn(\016)797 86 y Fj(0)816 79 y Fn(F)855 58 y Fi(\024)p Fj(0)848 91 y Fl(\013)913 79 y Fp(+)g Fn(\022)985 86 y Fj(0)1004 79 y Fn(F)1043 56 y Fi(\024)p Fj(0)1036 92 y Fl(\022)1101 79 y Fp(+)g Fn(\034)1171 86 y Fj(0)1191 79 y Fn(F)1230 58 y Fi(\024)p Fj(0)1223 91 y Fl(\034)1288 79 y Fp(+)1358 25 y Fj(6)1337 37 y Fk(X)1338 129 y Fl(i)p Fj(=1)1405 79 y Fn(g)1428 86 y Fl(i;)p Fj(0)1470 79 y Fn(F)1509 56 y Fi(\024)p Fj(0)1502 90 y Fl(i)1790 79 y Fp(\(58\))71 321 y Fn(\027)95 328 y Fj(0)129 321 y Fp(=)200 287 y Fn(Z)233 294 y Fj(0)p 186 309 81 2 v 186 355 a Fn(Z)219 362 y Fi(\000)p Fj(1)271 321 y Fn(n)300 328 y Fj(0)320 321 y Fn(;)57 b(\016)413 328 y Fj(0)446 321 y Fp(=)516 287 y Fn(Z)549 294 y Fj(0)p 503 309 V 503 355 a Fn(Z)536 362 y Fi(\000)p Fj(1)588 321 y Fp(\()p Fn(a)633 328 y Fj(0)664 321 y Fm(\000)10 b Fn(z)736 328 y Fj(0)756 321 y Fp(\))p Fn(;)56 b(\022)868 328 y Fj(0)902 321 y Fp(=)972 287 y Fn(Z)1005 294 y Fj(0)p 958 309 V 958 355 a Fn(Z)991 362 y Fi(\000)p Fj(1)1044 321 y Fn(i)1061 328 y Fj(0)1080 321 y Fn(;)8 b(\034)1123 328 y Fj(0)1157 321 y Fp(=)1227 287 y Fn(Z)1260 294 y Fj(0)p 1214 309 V 1214 355 a Fn(Z)1247 362 y Fi(\000)p Fj(1)1299 321 y Fn(t)1317 328 y Fj(0)1337 321 y Fn(;)56 b(g)1430 328 y Fl(i;)p Fj(0)1486 321 y Fp(=)1538 248 y Fk( )1589 287 y Fn(Z)1622 294 y Fj(0)p 1576 309 V 1576 355 a Fn(Z)1609 362 y Fi(\000)p Fj(1)1661 248 y Fk(!)1694 259 y Fj(2)1722 321 y Fn(\025)1750 328 y Fl(i;)p Fj(0)10 459 y Fp(It)16 b(is)g(con)o(v)o(enien)o(t)e(to)j(write)f(explicitely)d (the)j(propagator)i(asso)q(ciated)f(to)g(the)f(in)o(tegration)-13 537 y Fn(P)18 544 y Fl(Z)42 549 y Fh(\000)p Ff(1)86 537 y Fp(\()p Fn(d )164 519 y Fi(\024)p Fj(0)210 537 y Fp(\))h Fg(i.e.)g Fn(g)355 519 y Fj(0)353 550 y Fl(~)-18 b(!)r(;~)g(!)410 540 y Fh(0)423 537 y Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)602 502 y Fk(R)638 537 y Fn(P)669 544 y Fl(Z)693 549 y Fh(\000)p Ff(1)736 537 y Fp(\()p Fn(d )814 519 y Fj(0)834 537 y Fp(\))p Fn( )887 514 y Fj(+)p Fl(;)p Fj(0)885 551 y Fl(x;~)-18 b(!;\033)971 537 y Fn( )1005 514 y Fi(\000)p Fl(;)p Fj(0)1003 551 y Fl(y)q(;~)g(!)1054 541 y Fh(0)1065 551 y Fl(;\033)1098 537 y Fp(;)16 b(w)o(e)g(ha)o(v)o(e)f(that:)477 670 y Fn(g)502 649 y Fj(0)500 682 y Fl(~)-18 b(!)r(;~)g(!)557 672 y Fh(0)570 670 y Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)749 611 y Fk(Z)799 670 y Fn(dk)r(e)874 649 y Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))1000 670 y Fn(f)1024 677 y Fj(0)1044 670 y Fp(\()p Fn(k)1088 677 y Fj(0)1108 670 y Fn(;)1128 656 y(~)1130 670 y(k)r Fp(\))p Fn(T)1205 677 y Fi(\000)p Fj(1)1252 670 y Fp(\()p Fn(k)r Fp(\))1317 649 y Fi(\000)p Fj(1)1317 683 y Fl(~)-18 b(!)q(;~)g(!)1373 673 y Fh(0)1790 670 y Fp(\(59\))-13 797 y(where)136 925 y Fn(T)165 932 y Fi(\000)p Fj(1)212 925 y Fp(\()p Fn(k)r Fp(\))14 b(=)136 1047 y Fn(Z)169 1054 y Fi(\000)p Fj(1)217 1047 y Fp(\()p Fn(k)r Fp(\))290 961 y Fk(0)290 1036 y(@)347 1009 y Fn(\037)p Fp(\()394 996 y Fn(~)397 1009 y(k)f Fp(+)e Fn(p)508 1016 y Fl(F)537 1009 y Fp(\))556 991 y Fi(\000)p Fj(1)612 961 y Fk(\020)637 1009 y Fm(\000)p Fn(ik)718 1016 y Fj(0)748 1009 y Fp(+)795 996 y Fn(~)797 1009 y(k)824 991 y Fj(2)855 1009 y Fm(\000)g Fp(2)p Fn(\031)956 996 y(~)959 1009 y(k)986 961 y Fk(\021)1333 1009 y Fn(\033)1361 1016 y Fi(\000)p Fj(1)1408 1009 y Fp(\()p Fn(k)r Fp(\))609 1089 y Fn(\033)637 1096 y Fi(\000)p Fj(1)684 1089 y Fp(\()p Fn(k)r Fp(\))303 b Fn(\037)p Fp(\()p Fm(\000)1138 1076 y Fn(~)1141 1089 y(k)12 b Fp(+)f Fn(p)1251 1096 y Fl(F)1281 1089 y Fp(\))1300 1071 y Fi(\000)p Fj(1)1355 1041 y Fk(\020)1380 1089 y Fm(\000)p Fn(ik)1461 1096 y Fj(0)1492 1089 y Fp(+)1538 1076 y Fn(~)1541 1089 y(k)1568 1071 y Fj(2)1599 1089 y Fp(+)g(2)p Fn(\031)1699 1076 y(~)1702 1089 y(k)1729 1041 y Fk(\021)1774 961 y(1)1774 1036 y(A)-13 1216 y Fp(whic)o(h)k(is)h(w)o(ell)f(de\014ned)h(in)g(the)g(supp)q(ort)i(of)e Fn(f)843 1223 y Fj(0)863 1216 y Fp(\()p Fn(k)907 1223 y Fj(0)927 1216 y Fn(;)947 1203 y(~)949 1216 y(k)r Fp(\))g(and)134 1344 y Fn(A)171 1351 y Fi(\000)p Fj(1)218 1344 y Fp(\()p Fn(k)r Fp(\))e(=)g Fn(Z)382 1351 y Fi(\000)p Fj(1)429 1344 y Fp(\()p Fn(k)r Fp(\))502 1284 y Fk(\022)533 1296 y(\020)558 1344 y Fm(\000)p Fn(ik)639 1351 y Fj(0)669 1344 y Fp(+)716 1331 y Fn(~)718 1344 y(k)745 1324 y Fj(2)765 1296 y Fk(\021)790 1307 y Fj(2)821 1344 y Fm(\000)870 1296 y Fk(\020)895 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Fk(\021)994 1669 y Fm(\000)p Fn(\033)1061 1676 y Fi(\000)p Fj(1)1108 1669 y Fp(\()p Fn(k)r Fp(\))p Fn(\037)p Fp(\()1220 1656 y Fn(~)1223 1669 y(k)h Fp(+)f Fn(p)1333 1676 y Fl(F)1363 1669 y Fp(\))p Fn(\037)p Fp(\()p Fm(\000)1468 1656 y Fn(~)1471 1669 y(k)h Fp(+)f Fn(p)1581 1676 y Fl(F)1611 1669 y Fp(\))307 1748 y Fm(\000)p Fn(\033)374 1755 y Fi(\000)p Fj(1)420 1748 y Fp(\()p Fn(k)r Fp(\))p Fn(\037)p Fp(\()532 1735 y Fn(~)535 1748 y(k)i Fp(+)e Fn(p)646 1755 y Fl(F)676 1748 y Fp(\))p Fn(\037)p Fp(\()p Fm(\000)781 1735 y Fn(~)784 1748 y(k)h Fp(+)f Fn(p)894 1755 y Fl(F)924 1748 y Fp(\))41 b Fn(\037)p Fp(\()p Fm(\000)1070 1735 y Fn(~)1073 1748 y(k)13 b Fp(+)e Fn(p)1184 1755 y Fl(F)1214 1748 y Fp(\))1241 1700 y Fk(\020)1266 1748 y Fm(\000)p Fn(ik)1347 1755 y Fj(0)1377 1748 y Fp(+)1424 1735 y Fn(~)1426 1748 y(k)1453 1730 y Fj(2)1484 1748 y Fm(\000)g Fp(2)p Fn(\031)1585 1735 y(~)1588 1748 y(k)1615 1700 y Fk(\021)1660 1621 y(1)1660 1695 y(A)-13 1884 y Fp(W)l(e)16 b(p)q(erform)f(the)h(in)o(tegration)587 1849 y Fk(R)623 1884 y Fn(P)654 1891 y Fl(Z)678 1896 y Fh(\000)p Ff(1)722 1884 y Fp(\()p Fn(d )800 1866 y Fj(0)819 1884 y Fp(\))p Fn(e)861 1864 y Fi(\000)893 1856 y Fj(^)888 1864 y Fl(V)917 1853 y Ff(0)934 1864 y Fj(\()948 1831 y Fm(p)p 989 1831 112 2 v 989 1864 a Fl(Z)1013 1869 y Fh(\000)p Ff(1)1054 1864 y Fj(\()p Fl(k)q Fj(\))p Fl( )1125 1853 y Fh(\024)p Ff(0)1167 1864 y Fj(\))1196 1884 y Fp(=)e Fn(e)1271 1863 y Fi(\000)p Fl(V)1326 1852 y Fh(\000)p Ff(1)1368 1863 y Fj(\()1382 1831 y Fm(p)p 1423 1831 66 2 v 1423 1863 a Fl(Z)1447 1868 y Fh(\000)p Ff(1)1488 1863 y Fl( )1512 1852 y Fh(\024\000)p Ff(1)1578 1863 y Fj(\))1610 1884 y Fp(where)142 2012 y Fm(L)p Fn(V)216 1992 y Fi(\000)p Fj(1)263 2012 y Fp(\()p Fn( )316 1992 y Fi(\024\000)p Fj(1)390 2012 y Fp(\))41 b(=)h Fn(n)559 2019 y Fi(\000)p Fj(1)606 2012 y Fn(F)645 1992 y Fi(\024\000)p Fj(1)638 2025 y Fl(\027)730 2012 y Fp(+)11 b Fn(z)802 2019 y Fi(\000)p Fj(1)849 2012 y Fn(F)888 1992 y Fi(\024\000)p Fj(1)881 2025 y Fl(z)973 2012 y Fp(+)g Fn(a)1048 2019 y Fi(\000)p Fj(1)1095 2012 y Fn(F)1134 1992 y Fi(\024\000)p Fj(1)1127 2025 y Fl(\013)1219 2012 y Fp(+)g Fn(s)1291 2019 y Fi(\000)p Fj(1)1338 2012 y Fn(F)1377 1992 y Fi(\024\000)p Fj(1)1370 2025 y Fl(s)1462 2012 y Fp(+)g Fn(t)1529 2019 y Fj(0)1548 2012 y Fn(F)1587 1992 y Fi(\024\000)p Fj(1)1580 2025 y Fl(t)1672 2012 y Fp(+)450 2121 y(+)42 b Fn(i)547 2128 y Fi(\000)p Fj(1)594 2121 y Fn(F)633 2098 y Fi(\024\000)p Fj(1)626 2133 y Fl(\022)718 2121 y Fp(+)788 2067 y Fj(6)767 2079 y Fk(X)768 2170 y Fl(i)p Fj(=1)835 2121 y Fn(\025)863 2128 y Fl(i;)p Fi(\000)p Fj(1)932 2121 y Fn(F)971 2098 y Fi(\024\000)p Fj(1)964 2132 y Fl(i)1790 2121 y Fp(\(61\))-13 2263 y(and)17 b(of)f(course)g(the)g(pro)q(cedure)h(can)f(b)q(e)h (iterated.)10 2341 y(In)d(general)g(once)h(the)f(\014elds)g Fn( )584 2323 y Fj(0)603 2341 y Fn(;)8 b(:)g(:)g(:)f(;)h( )746 2323 y Fl(h)p Fj(+1)827 2341 y Fp(ha)o(v)o(e)14 b(b)q(een)g(in)o (tegrated)g(pro)q(ceding)g(as)h(ab)q(o)o(v)o(e,)g(w)o(e)e(ha)o(v)o(e) -13 2420 y(to)j(ev)m(alutate:)603 2498 y Fm(N)21 b Fp(=)717 2439 y Fk(Z)766 2498 y Fn(P)797 2505 y Fl(Z)821 2511 y Fe(h)844 2498 y Fp(\()p Fn(d )922 2477 y Fi(\024)p Fl(h)972 2498 y Fp(\))p Fn(e)1014 2477 y Fi(\000)p Fl(V)1069 2466 y Fe(h)1089 2477 y Fj(\()1103 2452 y Fi(p)p 1132 2452 45 2 v 25 x Fl(Z)1156 2483 y Fe(h)1176 2477 y Fl( )1200 2466 y Fh(\024)p Fe(h)1245 2477 y Fj(\))-13 2603 y Fp(where)16 b Fm(L)p Fn(V)201 2585 y Fl(h)224 2603 y Fp(\()p Fn( )277 2585 y Fi(\024)p Fl(h)326 2603 y Fp(\))g(as)h(de\014ned)f(is)g (de\014ned)g(as)907 2728 y(23)p eop %%Page: 24 24 24 23 bop 63 121 a Fm(L)p Fn(V)137 100 y Fl(h)159 121 y Fp(\()p Fn( )212 100 y Fi(\024)241 121 y Fp(\))14 b(=)g Fn(n)355 128 y Fl(h)377 121 y Fn(F)416 100 y Fi(\024)p Fl(h)409 133 y(\027)477 121 y Fp(+)d Fn(a)552 128 y Fl(h)574 121 y Fn(F)613 100 y Fi(\024)p Fl(h)606 133 y(\013)673 121 y Fp(+)g Fn(t)740 128 y Fl(h)762 121 y Fn(F)801 100 y Fi(\024)p 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Fl(\033)o(;~)g(!)485 564 y Fk(Z)535 623 y Fn(dk)r(C)622 630 y Fl(h)645 623 y Fp(\()p Fn(k)r Fp(\))p Fn(Z)743 630 y Fl(h)766 623 y Fp(\()p Fn(k)r Fp(\))831 562 y Fk(\024)857 589 y Fm(\000)p Fn(ik)938 596 y Fj(0)969 589 y Fp(+)1015 576 y Fn(~)1018 589 y(k)1045 571 y Fj(2)1076 589 y Fp(+)11 b(2)p Fn(~)-24 b(!)r(\031)1208 576 y(~)1211 589 y(k)p 857 611 381 2 v 927 665 a(\037)p Fp(\()p Fn(~)g(!)1006 652 y(~)1009 665 y(k)13 b Fp(+)e Fn(p)1120 672 y Fl(F)1149 665 y Fp(\))1243 623 y Fn( )1277 599 y Fj(+)p Fl(;)p Fi(\024)p Fl(h)1275 636 y(k)q Fj(+)p Fl(~)-18 b(!)q(\031)q(;~)g(!)r(;\033)1432 623 y Fn( )1466 599 y Fi(\000)p Fl(;)p Fi(\024)p Fl(h)1464 636 y(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)1632 623 y Fm(\000)264 764 y Fn(\033)292 771 y Fl(h)314 764 y Fp(\()p Fn(k)r Fp(\))p Fn( )413 740 y Fj(+)p Fl(;)p Fi(\024)p Fl(h)411 777 y(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)569 764 y Fn( )603 740 y Fi(\000)p Fl(;)p Fi(\024)p Fl(h)601 777 y(k)q Fi(\000)p Fl(~)g(!)q(\031)q(;)p Fi(\000)p Fl(~)g(!)r(;\033) 785 703 y Fk(\025\033)1790 764 y Fp(\(63\))-13 888 y(W)l(e)19 b(include)f(as)h(ab)q(o)o(v)o(e)h(in)e(the)h(Grassmanian)h(in)o (tegration)f(the)g(\\dangerous")i(terms)d(of)h Fm(L)p Fn(V)1783 870 y Fl(h)1805 888 y Fp(\()p Fn( )r Fp(\))-13 966 y Fg(i.e.)e Fn(s)94 973 y Fl(h)117 966 y Fn(F)156 948 y Fi(\024)p Fl(h)149 979 y(\033)221 966 y Fp(and)g Fn(z)339 973 y Fl(h)361 966 y Fn(F)400 944 y Fi(\024)p Fl(h)393 979 y(\020)449 966 y Fp(:)339 1064 y Fk(Z)389 1123 y Fn(P)420 1130 y Fl(Z)444 1136 y Fe(h)467 1123 y Fp(\()p Fn(d )545 1102 y Fi(\024)p Fl(h)595 1123 y Fp(\))p Fn(e)637 1102 y Fi(\000)p Fl(V)692 1091 y Fe(h)712 1102 y Fj(\()726 1077 y Fi(p)p 755 1077 45 2 v 25 x Fl(Z)779 1108 y Fe(h)799 1102 y Fl( )823 1091 y Fh(\024)p Fe(h)868 1102 y Fj(\))897 1123 y Fp(=)949 1064 y Fk(Z)999 1123 y Fn(P)1030 1130 y Fl(Z)1054 1136 y Fe(h)p Fh(\000)p Ff(1)1116 1123 y Fp(\()p Fn(d )1194 1102 y Fi(\024)p Fl(h)1243 1123 y Fp(\))p Fn(e)1285 1102 y Fi(\000)1317 1094 y Fj(~)1312 1102 y Fl(V)1341 1091 y Fe(h)1361 1102 y Fj(\()1375 1077 y Fi(p)p 1404 1077 V 25 x Fl(Z)1428 1108 y Fe(h)1448 1102 y Fl( )1472 1091 y Fh(\024)p Fe(h)1516 1102 y Fj(\))1790 1123 y Fp(\(64\))-13 1225 y(with)242 1345 y Fn(P)273 1352 y Fl(Z)297 1358 y Fe(h)p Fh(\000)p Ff(1)359 1345 y Fp(\()p Fn(d )437 1324 y Fi(\024)p Fl(h)486 1345 y Fp(\))d(=)586 1303 y Fk(Y)569 1396 y Fl(~)571 1405 y(k)q(;~)-18 b(!)q(;\033)662 1345 y Fn(d )721 1321 y Fj(+)p Fl(;)p Fi(\024)p Fl(h)719 1358 y(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)r(;\033)877 1345 y Fn(d )936 1321 y Fi(\000)p Fl(;)p Fi(\024)p Fl(h)934 1358 y(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)242 1514 y Fp(exp)316 1454 y Fk(\032)347 1514 y Fm(\000)394 1473 y Fk(X)398 1566 y Fl(\033)o(;~)g(!)463 1456 y Fk(Z)513 1514 y Fn(dk)r(C)600 1521 y Fl(h)622 1514 y Fp(\()p Fn(k)r Fp(\))p Fn(Z)720 1521 y Fl(h)p Fi(\000)p Fj(1)788 1514 y Fp(\()p Fn(k)r Fp(\))853 1454 y Fk(\024)880 1481 y Fm(\000)p Fn(ik)961 1488 y Fj(0)992 1481 y Fp(+)1038 1468 y Fn(~)1041 1481 y(k)1068 1463 y Fj(2)1098 1481 y Fp(+)11 b(2)p Fn(~)-24 b(!)s(\031)1231 1468 y(~)1234 1481 y(k)p 880 1503 381 2 v 949 1556 a(\037)p Fp(\()p Fn(~)g(!)1029 1543 y(~)1031 1556 y(k)13 b Fp(+)e Fn(p)1142 1563 y Fl(F)1172 1556 y Fp(\))1265 1514 y Fn( )1299 1491 y Fj(+)p Fl(;)p Fi(\024)p Fl(h)1297 1528 y(k)q Fj(+)p Fl(~)-18 b(!)r(\031)q(;~)g(!)q(;\033)1455 1514 y Fn( )1489 1491 y Fi(\000)p Fl(;)p Fi(\024)p Fl(h)1487 1528 y(k)q Fj(+)p Fl(~)g(!)q(\031)q(;~)g(!)q(;\033)1655 1514 y Fm(\000)242 1655 y Fn(\033)270 1662 y Fl(h)p Fi(\000)p Fj(1)337 1655 y Fp(\()p Fn(k)r Fp(\))p Fn( )436 1632 y Fj(+)p Fl(;)p Fi(\024)p Fl(h)434 1669 y(k)q Fj(+)p Fl(~)g(!)r(\031)q(;~)g(!)q(;\033)591 1655 y Fn( )625 1632 y Fi(\000)p Fl(;)p Fi(\024)p Fl(h)623 1669 y(k)q Fi(\000)p Fl(~)g(!)r(\031)q(;)p Fi(\000)p Fl(~)g(!)r(;\033)808 1595 y Fk(\025\033)1790 1655 y Fp(\(65\))-13 1781 y(with)18 b Fn(Z)133 1788 y Fl(h)p Fi(\000)p Fj(1)201 1781 y Fp(\()p Fn(k)r Fp(\))g(=)h Fn(Z)374 1788 y Fl(h)396 1781 y Fp(\()p Fn(k)r Fp(\))13 b(+)g Fn(C)564 1760 y Fi(\000)p Fj(1)560 1793 y Fl(h)611 1781 y Fp(\()p Fn(k)r Fp(\))p Fn(Z)709 1788 y Fl(h)731 1781 y Fp(\()p Fn(k)r Fp(\))p Fn(z)819 1788 y Fl(h)841 1781 y Fp(,)38 b Fn(Z)926 1788 y Fl(h)p Fi(\000)p Fj(1)993 1781 y Fp(\()p Fn(k)r Fp(\))p Fn(\033)1086 1788 y Fl(h)p Fi(\000)p Fj(1)1172 1781 y Fp(=)18 b Fn(Z)1261 1788 y Fl(h)1284 1781 y Fp(\()p Fn(k)r Fp(\))p Fn(\033)1377 1788 y Fl(h)1412 1781 y Fp(+)12 b Fn(Z)1495 1788 y Fl(h)1518 1781 y Fp(\()p Fn(k)r Fp(\))p Fn(C)1622 1760 y Fi(\000)p Fj(1)1618 1793 y Fl(h)1669 1781 y Fp(\()p Fn(k)r Fp(\))p Fn(s)1757 1788 y Fl(h)1798 1781 y Fp(and)-6 1846 y(~)-13 1859 y Fn(V)26 1841 y Fl(h)62 1859 y Fp(=)i Fm(L)155 1846 y Fp(~)148 1859 y Fn(V)188 1841 y Fl(h)221 1859 y Fp(+)d(\(1)h Fm(\000)e(L)p Fp(\))p Fn(V)467 1841 y Fl(h)506 1859 y Fp(with)224 1979 y Fm(L)265 1966 y Fp(~)258 1979 y Fn(V)297 1958 y Fl(h)320 1979 y Fp(\()p Fn( )373 1958 y Fi(\024)p Fl(h)422 1979 y Fp(\))41 b(=)h Fn(n)591 1986 y Fl(h)613 1979 y Fn(F)652 1958 y Fi(\024)p Fl(h)645 1991 y(\027)713 1979 y Fp(+)11 b(\()p Fn(a)807 1986 y Fl(h)840 1979 y Fm(\000)g Fn(z)913 1986 y Fl(h)935 1979 y Fp(\))p Fn(F)986 1986 y Fl(\013)1021 1979 y Fp(+)g Fn(t)1088 1986 y Fl(h)1110 1979 y Fn(F)1142 1986 y Fl(t)1167 1979 y Fp(+)g Fn(i)1233 1986 y Fl(h)1255 1979 y Fn(F)1287 1986 y Fl(\022)1318 1979 y Fp(+)482 2070 y(+)42 b Fn(\025)590 2077 y Fj(1)p Fl(;h)640 2070 y Fn(F)672 2077 y Fj(1)703 2070 y Fp(+)11 b Fn(\025)780 2077 y Fj(2)p Fl(;h)830 2070 y Fn(F)862 2077 y Fj(2)892 2070 y Fp(+)g Fn(\025)969 2077 y Fj(3)p Fl(;h)1019 2070 y Fn(F)1051 2077 y Fj(3)1082 2070 y Fp(+)g Fn(\025)1159 2077 y Fj(4)p Fl(;h)1209 2070 y Fn(F)1241 2077 y Fj(4)1272 2070 y Fp(+)g Fn(\025)1349 2077 y Fj(5)p Fl(;h)1399 2070 y Fn(F)1431 2077 y Fj(5)1461 2070 y Fp(+)g Fn(\025)1538 2077 y Fj(6)p Fl(;h)1589 2070 y Fn(F)1621 2077 y Fj(6)1790 2070 y Fp(\(66\))10 2190 y(Finall)16 b(w)o(e)f(write:)388 2309 y Fm(N)21 b Fp(=)502 2251 y Fk(Z)552 2309 y Fn(P)583 2316 y Fl(Z)607 2322 y Fe(h)p Fh(\000)p Ff(1)668 2309 y Fp(\()p Fn(d )746 2289 y Fi(\024)p Fl(h)p Fi(\000)p Fj(1)841 2309 y Fp(\))868 2251 y Fk(Z)918 2309 y Fn(P)949 2316 y Fl(Z)973 2322 y Fe(h)p Fh(\000)p Ff(1)1035 2309 y Fp(\()p Fn(d )1113 2289 y Fl(h)1135 2309 y Fp(\))p Fn(e)1177 2288 y Fi(\000)1209 2280 y Fj(^)1204 2288 y Fl(V)1233 2276 y Fe(h)1253 2288 y Fj(\()1267 2256 y Fm(p)p 1308 2256 84 2 v 1308 2288 a Fl(Z)1332 2294 y Fe(h)p Fh(\000)p Ff(1)1391 2288 y Fl( )1415 2276 y Fh(\024)p Fe(h)1460 2288 y Fj(\))-13 2429 y Fp(with)170 2586 y Fn(g)195 2565 y Fl(h)193 2598 y(~)-18 b(!)r(;~)g(!)250 2589 y Fh(0)263 2586 y Fp(\()p Fn(x)11 b Fm(\000)g Fn(y)r Fp(\))i(=)481 2527 y Fk(Z)531 2586 y Fn(P)562 2593 y Fl(Z)586 2599 y Fe(h)p Fh(\000)p Ff(1)647 2586 y Fp(\()p Fn(d )725 2565 y Fl(h)748 2586 y Fp(\))p Fn( )801 2562 y Fj(+)p Fl(;h)799 2599 y(x;~)-18 b(!;\033)884 2586 y Fn( )918 2562 y Fi(\000)p Fl(;h)916 2599 y(y)q(;~)g(!)967 2590 y Fh(0)979 2599 y Fl(;\033)1026 2586 y Fp(=)1078 2527 y Fk(Z)1128 2586 y Fn(dk)r(e)1203 2565 y Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))1329 2586 y Fn(f)1353 2593 y Fl(h)1376 2586 y Fp(\()p Fn(k)1420 2593 y Fj(0)1440 2586 y Fn(;)1460 2573 y(~)1462 2586 y(k)r Fp(\))p Fn(T)1537 2593 y Fl(h)1559 2586 y Fp(\()p Fn(k)r Fp(\))1624 2565 y Fi(\000)p Fj(1)1624 2599 y Fl(~)g(!)q(;~)g(!) 1680 2590 y Fh(0)1790 2586 y Fp(\(67\))907 2728 y(24)p eop %%Page: 25 25 25 24 bop -13 -5 a Fp(where)171 112 y Fn(A)208 119 y Fl(h)230 112 y Fp(\()p Fn(k)r Fp(\))14 b(=)g Fn(Z)394 119 y Fl(h)417 112 y Fp(\()p Fn(k)r Fp(\))490 52 y Fk(\022)521 64 y(\020)545 112 y Fm(\000)p Fn(ik)626 119 y Fj(0)657 112 y Fp(+)703 99 y Fn(~)706 112 y(k)733 92 y Fj(2)753 64 y Fk(\021)777 76 y Fj(2)808 112 y Fm(\000)858 64 y Fk(\020)883 112 y Fp(2)p Fn(\031)934 99 y(~)937 112 y(k)964 64 y Fk(\021)988 76 y Fj(2)1019 112 y Fm(\000)d Fn(\037)p Fp(\()1116 99 y Fn(~)1119 112 y(k)i Fp(+)e Fn(p)1230 119 y Fl(F)1260 112 y Fp(\))p Fn(\037)p Fp(\()p Fm(\000)1365 99 y Fn(~)1368 112 y(k)h Fp(+)f Fn(p)1478 119 y Fl(F)1508 112 y Fp(\))p 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y(k)i Fp(+)e Fn(p)634 551 y Fl(F)663 544 y Fp(\))p Fn(\037)p Fp(\()p Fm(\000)768 531 y Fn(~)771 544 y(k)i Fp(+)e Fn(p)882 551 y Fl(F)912 544 y Fp(\))43 b Fn(\037)p Fp(\()p Fm(\000)1060 531 y Fn(~)1063 544 y(k)13 b Fp(+)e Fn(p)1174 551 y Fl(F)1203 544 y Fp(\))1230 496 y Fk(\020)1255 544 y Fm(\000)p Fn(ik)1336 551 y Fj(0)1367 544 y Fp(+)1413 531 y Fn(~)1416 544 y(k)1443 526 y Fj(2)1474 544 y Fm(\000)g Fp(2)p Fn(\031)1575 531 y(~)1578 544 y(k)1604 496 y Fk(\021)1650 416 y(1)1650 491 y(A)-13 646 y Fp(and)383 730 y Fm(L)424 717 y Fp(^)417 730 y Fn(V)456 709 y Fl(h)479 730 y Fp(\()p Fn( )r Fp(\))i(=)h Fn(\015)644 709 y Fl(h)666 730 y Fn(\027)690 737 y Fl(h)713 730 y Fn(F)745 737 y Fl(\027)777 730 y Fp(+)d Fn(\016)848 737 y Fl(h)870 730 y Fn(F)902 737 y Fl(\013)937 730 y Fp(+)g Fn(\022)1009 737 y Fl(h)1031 730 y Fn(F)1063 737 y Fl(\022)1093 730 y Fp(+)h Fn(\034)1164 737 y Fl(h)1186 730 y Fn(F)1218 737 y Fl(\034)1250 730 y Fp(+)1320 676 y Fj(6)1299 688 y Fk(X)1301 779 y Fl(i)p Fj(=1)1368 730 y 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Fn(Z)936 1034 y Fl(h)p 880 1049 V 880 1095 a Fn(Z)913 1102 y Fl(h)p Fi(\000)p Fj(1)986 988 y Fk(!)1019 999 y Fj(2)1047 1061 y Fn(\025)1075 1068 y Fl(i;)p Fj(0)1790 1061 y Fp(\(69\))10 1191 y(In)17 b(the)g(ab)q(o)o(v)o(e)f(form)o(ulae)f(w)o(e)i(ha)o(v)o(e) f(used)h(that)g Fn(\033)928 1198 y Fl(h)950 1191 y Fp(\()p Fn(k)r Fp(\))p Fn(;)8 b(Z)1070 1198 y Fl(h)1093 1191 y Fp(\()p Fn(k)r Fp(\))17 b(do)g(not)g(dep)q(end)g(on)h Fn(~)-25 b(!)s Fp(:)22 b(this)17 b(follo)o(ws)-13 1269 y(from)10 b(the)h("rotation")i(in)o(v)m(ariance)d(of)i(the)f(theory)l (,)h Fg(i.e.)17 b Fp(in)o(v)m(ariance)11 b(under)g(transformation)g Fn(~)-24 b(x)14 b Fm(!)g(\000)o Fn(~)-23 b(x)o Fp(,)-13 1347 y(and)17 b(from)e(the)h(de\014nition)g(of)g Fn(C)592 1327 y Fi(\000)p Fj(1)588 1360 y Fl(h)639 1347 y Fp(.)10 1426 y(In)h(the)f(follo)o(wing)g(w)o(e)h(call)e Fn(Z)557 1433 y Fl(h)595 1426 y Fp(=)f Fn(Z)680 1433 y Fl(h)703 1426 y Fp(\()p Fn(\015)750 1408 y Fl(h)772 1426 y Fp(\))j(and)g Fn(\033)931 1433 y Fl(h)967 1426 y Fp(=)e Fn(\033)1048 1433 y Fl(h)1070 1426 y Fp(\()p Fn(\015)1117 1408 y Fl(h)1139 1426 y Fp(\).)22 b(Note)17 b(moreo)o(v)o(er)d(that)j(if)f Fn(k)j Fp(b)q(elongs)-13 1504 y(to)d(the)g(supp)q(ort)i(of)e Fn(f)390 1511 y Fl(h)413 1504 y Fp(\()429 1491 y Fn(~)432 1504 y(k)r(;)8 b(k)506 1511 y Fj(0)526 1504 y Fp(\))16 b(w)o(e)g(ha)o(v)o(e)f(that:)200 1621 y Fn(Z)233 1628 y Fl(h)256 1621 y Fp(\()p Fn(k)r Fp(\))f(=)g Fn(Z)420 1628 y Fj(0)500 1579 y Fk(Y)448 1672 y Fl(h)p Fj(+1)p Fi(\025)p Fl(k)q(>)p Fj(0)605 1621 y Fp(\(1)e(+)f Fn(C)748 1600 y Fi(\000)p Fj(1)744 1633 y Fl(k)794 1621 y Fp(\()p Fn(k)r Fp(\))p Fn(z)882 1628 y Fl(k)903 1621 y Fp(\))j(=)g Fn(Z)1021 1628 y Fj(0)1041 1621 y Fp(\(1)d(+)g Fn(C)1183 1600 y Fi(\000)p Fj(1)1179 1633 y Fl(h)p Fj(+1)1247 1621 y Fp(\()p Fn(k)r Fp(\)\))p Fn(z)1354 1628 y Fl(h)p Fj(+1)1481 1579 y Fk(Y)1429 1672 y Fl(h)p Fj(+2)p Fi(\025)p Fl(k)q(>)p Fj(0)1595 1621 y Fp(2)p Fn(z)1642 1628 y Fl(k)1790 1621 y Fp(\(70\))-13 1761 y(A)16 b(similar)e(form)o(ula)h(holds)h(also)h (for)g Fn(\033)707 1768 y Fl(h)729 1761 y Fp(\()p Fn(k)r Fp(\).)10 1840 y(In)g(order)h(to)f(study)g(the)g(renormalization)f (group)i(\015o)o(w)f(w)o(e)g(need)g(estimates)e(on)j(the)f(propagator,) -13 1918 y(whic)o(h)e(are)i(giv)o(en)e(b)o(y)h(the)g(follo)o(wing)g (lemm)o(a)e(that)j(will)e(b)q(e)h(pro)o(v)o(ed)g(in)f(app)q(endix)i(3:) -13 2028 y Fd(Lemm)o(a)f(2.1)24 b Fg(The)18 b(pr)n(op)n(agator)e Fn(g)648 2010 y Fl(h)646 2041 y(~)-18 b(!)q(;~)g(!)704 2028 y Fp(\()p Fn(x)11 b Fm(\000)g Fn(y)r Fp(\))17 b Fg(c)n(an)g(b)n(e)h(written)h(as)357 2145 y Fn(g)382 2125 y Fl(h)380 2157 y(~)-18 b(!)q(;~)g(!)438 2145 y Fp(\()p Fn(x)11 b Fm(\000)g Fn(y)r Fp(\))i(=)h Fn(g)681 2125 y Fl(h)679 2157 y(~)-18 b(!)q(;L)738 2145 y Fp(\()p Fn(x)11 b Fm(\000)g Fn(y)r Fp(\))f(+)h Fn(C)985 2152 y Fj(1)1005 2145 y Fp(\()p Fn(x)g Fm(\000)f Fn(y)r Fp(\))h(+)g Fn(C)1252 2152 y Fj(2)1272 2145 y Fp(\()p Fn(x)f Fm(\000)h Fn(y)r Fp(\))366 b(\(71\))502 2263 y Fn(g)527 2242 y Fl(h)525 2275 y(~)-18 b(!)r(;L)585 2263 y Fp(\()p Fn(x)11 b Fm(\000)f Fn(y)r Fp(\))k(=)802 2204 y Fk(Z)887 2229 y Fn(dk)p 857 2251 112 2 v 857 2297 a Fp(\(2)p Fn(\031)r Fp(\))949 2283 y Fj(2)979 2229 y Fn(e)1002 2211 y Fl(ik)q(x)p 979 2251 76 2 v 989 2297 a Fn(Z)1022 2304 y Fl(h)1064 2229 y Fn(f)5 b Fp(\()p Fn(\015)1140 2211 y Fi(\000)p Fj(2)p Fl(h)1208 2229 y Fp(\()p Fn(k)1254 2211 y Fj(2)1252 2241 y(0)1285 2229 y Fp(+)1332 2216 y Fn(~)1334 2229 y(k)1361 2211 y Fj(2)1381 2229 y Fp(\)\))p 1064 2251 355 2 v 1132 2305 a Fm(\000)p Fn(ik)1213 2312 y Fj(0)1243 2305 y Fp(+)11 b Fn(~)-24 b(!)1322 2292 y(~)1324 2305 y(k)-13 2413 y Fg(wher)n(e)17 b Fm(j)p Fn(C)173 2420 y Fj(1)193 2413 y Fp(\()p Fn(x)11 b Fm(\000)f Fn(y)r Fp(\))p Fm(j)k(\024)525 2392 y Fl(\015)545 2380 y Ff(2)p Fe(h)580 2392 y Fl(C)605 2398 y Fe(N)p 430 2402 299 2 v 430 2430 a Fl(Z)454 2436 y Fe(h)474 2430 y Fj(\(1+\()p Fl(\015)567 2421 y Fe(h)587 2430 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)q Fi(j)p Fj(\))686 2421 y Fe(N)715 2430 y Fj(\))751 2413 y Fg(and)k Fm(j)p Fn(C)895 2420 y Fj(2)914 2413 y Fp(\()p Fn(x)11 b Fm(\000)g Fn(y)r Fp(\))p Fm(j)i(\024)1154 2392 y Fl(\015)1174 2380 y Fe(h)p 1152 2402 45 2 v 1152 2430 a Fl(Z)1176 2436 y Fe(h)1209 2365 y Fk(\020)1239 2392 y Fl(\033)1259 2398 y Fe(h)p 1239 2402 41 2 v 1239 2430 a Fl(\015)1259 2421 y Fe(h)1284 2365 y Fk(\021)1309 2377 y Fj(2)1435 2393 y Fl(C)1460 2399 y Fe(N)p 1342 2402 241 2 v 1342 2430 a Fj(\(1+\()p Fl(\015)1435 2421 y Fe(h)1455 2430 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)q Fi(j)p Fj(\))1554 2421 y Fe(N)1588 2413 y Fg(.)22 b(Mor)n(e)n(over)512 2556 y Fm(j)p Fn(g)551 2536 y Fl(h)549 2569 y(~)-18 b(!)q(;)p Fi(\000)p Fl(~)g(!)635 2556 y Fp(\()p Fn(x)10 b Fm(\000)h Fn(y)r Fp(\))p Fm(j)i(\024)875 2523 y Fn(\015)903 2505 y Fl(h)p 872 2545 56 2 v 872 2591 a Fn(Z)905 2598 y Fl(h)938 2523 y Fn(\033)966 2530 y Fl(h)p 938 2545 51 2 v 938 2591 a Fn(\015)966 2576 y Fl(h)1138 2523 y Fn(C)1173 2530 y Fl(N)p 998 2545 349 2 v 998 2591 a Fp(1)f(+)f(\()p Fn(\015)1130 2576 y Fl(h)1152 2591 y Fm(j)p Fn(x)g Fm(\000)f Fn(y)r Fm(j)p Fp(\))1313 2576 y Fl(N)907 2728 y Fp(25)p eop %%Page: 26 26 26 25 bop -13 -5 a Fp(W)l(e)16 b(can)g(describ)q(e)g(the)g(structure)g (of)705 -17 y(^)697 -5 y Fn(V)737 -23 y Fl(h)775 -5 y Fp(in)g(term)f(of)h(a)h Fg(tr)n(e)n(e)g(exp)n(ansion)p Fp(.)22 b(Note)16 b(in)g(fact)g(that:)445 123 y Fn(e)468 103 y Fi(\000)p Fl(V)523 91 y Fe(h)543 103 y Fj(\()557 77 y Fi(p)p 586 77 45 2 v 26 x Fl(Z)610 109 y Fe(h)630 103 y Fl( )654 91 y Fh(\024)p Fe(h)699 103 y Fj(\))728 123 y Fp(=)780 65 y Fk(Z)830 123 y Fn(P)861 130 y Fl(Z)885 136 y Fe(h)908 123 y Fp(\()p Fn(d )986 103 y Fl(h)p Fj(+1)1053 123 y Fp(\))p Fn(e)1095 103 y Fi(\000)1127 94 y Fj(^)1122 103 y Fl(V)1150 91 y Fe(h)p Ff(+1)1209 103 y Fj(\()1223 77 y Fi(p)p 1252 77 V 26 x Fl(Z)1276 109 y Fe(h)1296 103 y Fl( )1320 91 y Fh(\024)p Fe(h)p Ff(+1)1403 103 y Fj(\))-13 251 y Fp(and)h(at)f(eac)o(h)g(step)g Fn(V)394 233 y Fl(h)416 251 y Fp(\()435 213 y Fm(p)p 476 213 56 2 v 476 251 a Fn(Z)509 258 y Fl(h)532 251 y Fn( )r Fp(\))g(is)g (written)g(as)440 381 y(^)433 394 y Fn(V)472 373 y Fl(h)494 394 y Fp(\()p Fn( )r Fp(\))e(=)f Fm(L)665 373 y Fi(\003)685 394 y Fn(V)725 373 y Fl(h)755 321 y Fk( )816 322 y Fm(p)p 857 322 V 857 360 a Fn(Z)890 367 y Fl(h)p 793 382 143 2 v 793 391 a Fm(p)p 835 391 101 2 v 37 x Fn(Z)868 435 y Fl(h)p Fi(\000)p Fj(1)941 394 y Fn( )975 321 y Fk(!)1018 394 y Fp(+)e Fm(R)p Fn(V)1149 373 y Fl(h)1179 321 y Fk( )1240 322 y Fm(p)p 1281 322 56 2 v 1281 360 a Fn(Z)1314 367 y Fl(h)p 1217 382 143 2 v 1217 391 a Fm(p)p 1259 391 101 2 v 37 x Fn(Z)1292 435 y Fl(h)p Fi(\000)p Fj(1)1364 394 y Fn( )1398 321 y Fk(!)-13 539 y Fp(where)17 b Fm(L)163 521 y Fi(\003)183 539 y Fn(V)222 521 y Fl(h)262 539 y Fp(di\013ers)g(from)g Fm(L)p Fn(V)598 521 y Fl(h)638 539 y Fp(only)g(b)q(ecause)h(it)f(do)q(es)h(not)g(con)o(tain)f(an)o (ymore)f(the)h(addends)h Fn(F)1857 546 y Fl(z)-13 617 y Fp(and)f Fn(F)114 624 y Fl(\033)137 617 y Fp(.)k(W)l(e)16 b(can)g(write)399 751 y Fn(V)439 730 y Fl(k)460 751 y Fp(\()p Fn( )513 730 y Fi(\024)p Fl(k)561 751 y Fp(\))e(=)662 697 y Fi(1)649 709 y Fk(X)646 800 y Fl(n)p Fj(=1)731 709 y Fk(X)721 797 y Fl(\034)t Fi(2)p Fl(\034)781 801 y Fe(n)809 751 y Fn(V)848 730 y Fl(k)870 751 y Fp(\()p Fn(\034)s(;)8 b( )969 730 y Fi(\024)p Fl(k)1017 751 y Fp(\))399 888 y Fn(V)439 867 y Fl(k)460 888 y Fp(\()p Fn(\034)s(;)g(Z)567 844 y Ff(1)p 567 850 16 2 v 567 871 a(2)558 900 y Fl(k)589 888 y Fn( )623 867 y Fi(\024)p Fl(k)671 888 y Fp(\))14 b(=)g Fm(O)826 854 y Fp(1)p 802 876 74 2 v 802 922 a Fn(s)825 929 y Fl(v)842 934 y Ff(0)862 922 y Fp(!)880 888 y Fn(E)919 867 y Fl(T)916 900 y(k)q Fj(+1)983 888 y Fp([)1004 875 y(^)997 888 y Fn(V)1035 867 y Fl(k)q Fj(+1)1102 888 y Fp(\()p Fn(\034)1148 867 y Fj(1)1167 888 y Fn(;)8 b(Z)1231 844 y Ff(1)p 1231 850 16 2 v 1231 871 a(2)1222 900 y Fl(k)q Fj(+1)1289 888 y Fn( )1323 867 y Fi(\024)p Fl(k)q Fj(+1)1416 888 y Fp(\))p Fn(;)g(:)g(:)g(:)o Fp(\)])243 b(\(72\))-13 1028 y(where)14 b Fn(n)g Fm(\025)g Fp(2,)g Fn(\034)301 1009 y Fj(1)329 1028 y Fn(:)8 b(:)g(:)g(\034)422 1009 y Fl(s)438 1013 y Fe(v)454 1020 y Ff(0)489 1028 y Fp(are)15 b(the)f(subtrees)h (starting)g(from)e Fn(v)1157 1035 y Fj(0)1191 1028 y Fp(\(the)h(\014rst)h(v)o(ertex)e(ab)q(o)o(v)o(e)i(the)f(ro)q(ot\),)-13 1106 y(the)19 b(sym)o(b)q(ols)f Fn(E)299 1113 y Fl(h)321 1106 y Fp(,)i Fn(E)394 1088 y Fl(T)391 1118 y(h)440 1106 y Fp(denote)g(the)e(exp)q(ectations)h(with)g(resp)q(ect)g(to)h(a)f (grassmanian)h(in)o(tegration)-13 1184 y(with)e(propagator)j Fn(g)377 1159 y Fj(\()p Fl(h)p Fj(\))375 1197 y Fl(~)-18 b(!)397 1202 y Ff(1)415 1197 y Fl(;~)g(!)447 1202 y Ff(2)485 1184 y Fp(and)19 b Fm(O)h Fp(is)e(equal)g(to)h Fm(L)920 1166 y Fi(\003)940 1184 y Fp(,)g(if)f(the)h(tree)f(con)o(tributes)g(to) h(the)g(lo)q(cal)f(part)h(of)-13 1262 y(the)f(p)q(oten)o(tial,)h(or)g Fm(R)p Fp(,)h(if)e(it)g(con)o(tributes)g(to)i(the)e(irrelev)m(an)o(t)g (part.)29 b(The)19 b(trees)f(are)h(de\014ned)g(in)f(an)-13 1341 y(analogous)g(w)o(a)o(y)e(as)h(in)f(the)g(preceding)f(section.)-13 1507 y Fc(2.3)69 b(The)23 b(\015o)n(w)g(of)g(the)g(renormali)o(zati)o (on)e(group)-13 1618 y Fp(F)l(rom)16 b(the)i(ab)q(o)o(v)o(e)g (discussion)g(w)o(e)f(ha)o(v)o(e)h(that)g(the)f(set)h Fn(v)1054 1625 y Fl(j;h)p Fi(\000)p Fj(1)1163 1618 y Fp(is)g(related)f(to)h(the)g Fn(v)1549 1625 y Fl(j;h)1593 1616 y Fh(0)1624 1618 y Fp(with)g Fn(h)1765 1600 y Fi(0)1793 1618 y Fm(\025)e Fn(h)-13 1696 y Fp(b)o(y)g(the)g(relation:)512 1824 y(\026)-26 b Fn(v)534 1831 y Fl(i;h)p Fi(\000)p Fj(1)637 1824 y Fp(=)14 b(\()735 1790 y Fn(Z)768 1797 y Fl(h)p 713 1812 101 2 v 713 1858 a Fn(Z)746 1865 y Fl(h)p Fi(\000)p Fj(1)819 1824 y Fp(\))838 1804 y Fl(\013)861 1809 y Fe(i)876 1824 y Fp([)r(\026)-26 b Fn(v)914 1831 y Fl(i;h)968 1824 y Fp(+)11 b Fn(\014)1048 1804 y Fl(h)1045 1836 y(g)q(;i)1086 1824 y Fp(\()p Fm(f)p Fn(v)1154 1831 y Fl(h)1176 1824 y Fm(g)p Fn(;)d(:)g(:)g(:)g(;)g Fm(f)p Fn(v)1360 1831 y Fj(0)1379 1824 y Fm(g)p Fp(\)])510 1954 y Fn(\033)538 1961 y Fl(h)p Fi(\000)p Fj(1)619 1954 y Fp(=)699 1920 y Fn(Z)732 1927 y Fl(h)p 676 1942 V 676 1988 a Fn(Z)709 1995 y Fl(h)p Fi(\000)p Fj(1)782 1954 y Fp([)p Fn(\033)824 1961 y Fl(h)857 1954 y Fp(+)j Fn(\014)937 1933 y Fl(h)934 1966 y(\033)958 1954 y Fp(\()p Fm(f)p Fn(v)1026 1961 y Fl(h)1048 1954 y Fm(g)p Fn(;)d(:)g(:)g(:)g(;)g Fm(f)p Fn(v)1232 1961 y Fj(0)1251 1954 y Fm(g)p Fp(\)])481 b(\(73\))510 2083 y Fn(\027)534 2090 y Fl(h)p Fi(\000)p Fj(1)616 2083 y Fp(=)695 2049 y Fn(Z)728 2056 y Fl(h)p 673 2071 V 673 2117 a Fn(Z)706 2124 y Fl(h)p Fi(\000)p Fj(1)778 2083 y Fp([)p Fn(\015)s(\027)844 2090 y Fl(h)878 2083 y Fp(+)11 b Fn(\014)958 2062 y Fl(h)955 2095 y(\033)979 2083 y Fp(\()p Fm(f)p Fn(v)1047 2090 y Fl(h)1069 2083 y Fm(g)p Fn(;)d(:)g(:)g(:)f(;)h Fm(f)p Fn(v)1252 2090 y Fj(0)1271 2083 y Fm(g)p Fp(\)])510 2212 y(1)15 b(=)628 2179 y Fn(Z)661 2186 y Fl(h)p 605 2201 V 605 2247 a Fn(Z)638 2254 y Fl(h)p Fi(\000)p Fj(1)711 2212 y Fp([1)c(+)g Fn(\014)840 2192 y Fl(h)837 2225 y(z)862 2212 y Fp(\()p Fm(f)p Fn(v)930 2219 y Fl(h)952 2212 y Fm(g)p Fn(;)d(:)g(:)g(:)f(;)h Fm(f)p Fn(v)1135 2219 y Fj(0)1154 2212 y Fm(g)p Fp(\)])-13 2349 y(where)20 b Fn(v)156 2356 y Fl(h;i)221 2349 y Fp(=)i Fm(f)r Fp(\026)-26 b Fn(v)330 2356 y Fl(h;i)373 2349 y Fn(;)8 b(\027)419 2356 y Fl(h)442 2349 y Fn(;)g(Z)497 2356 y Fl(h)519 2349 y Fn(;)g(\033)569 2356 y Fl(h)591 2349 y Fm(g)p Fp(,)21 b Fn(\013)682 2356 y Fl(i)718 2349 y Fp(=)g(2)g(for)g(the)g Fn(g)1013 2356 y Fl(i;h)1078 2349 y Fp(and)g Fn(\013)1208 2356 y Fl(i)1244 2349 y Fp(=)g(1)g(otherwise.)34 b(The)21 b(function)-13 2427 y Fn(\014)18 2409 y Fl(h)15 2439 y(i)40 2427 y Fp(\()p Fm(f)p Fn(v)108 2434 y Fl(h)129 2427 y Fm(g)p Fn(;)8 b(:)g(:)g(:)g(;)g Fm(f)p Fn(v)313 2434 y Fj(0)332 2427 y Fm(g)p Fp(\))i(is)h(called)f Fg(Beta)j(function)f Fp(and)g(from)d (eq.\(72\))i(w)o(e)f(kno)o(w)h(that)h Fn(\014)1536 2409 y Fl(h)1533 2439 y(i)1557 2427 y Fp(\()p Fm(f)p Fn(v)1625 2434 y Fl(h)1647 2427 y Fm(g)p Fn(;)c(:)g(:)g(:)f(;)h Fm(f)p Fn(v)1830 2434 y Fj(0)1850 2427 y Fm(g)p Fp(\))-13 2505 y(=)47 2472 y Fk(P)91 2485 y Fi(1)91 2518 y Fl(n)p Fj(=0)168 2505 y Fn(\014)199 2482 y Fl(h;n)196 2517 y(i)252 2505 y Fp(\()p Fm(f)p Fn(v)320 2512 y Fl(h)342 2505 y Fm(g)p Fn(;)g(:)g(:)g(:)f(;)h Fm(f)p Fn(v)525 2512 y Fj(0)544 2505 y Fm(g)p Fp(\).)36 b(Noting)21 b(that)h Fm(j)p Fn(g)952 2512 y Fl(~)-18 b(!)q(;~)g(!)1008 2503 y Fh(0)1022 2505 y Fp(\()p Fn(x)14 b Fm(\000)g Fn(y)r Fp(\))p Fm(j)22 b(\024)1349 2484 y Fl(C)1374 2490 y Fe(N)1403 2484 y Fl(\015)1423 2472 y Fe(h)p 1283 2494 227 2 v 1283 2523 a Fj(1+)p Fl(\015)1348 2513 y Fe(N)s(h)1395 2523 y Fi(j)p Fl(x)p Fi(\000)p Fl(y)q Fi(j)1480 2513 y Fe(N)1523 2457 y Fk(\020)1553 2485 y Fl(\033)1573 2491 y Fe(h)p 1553 2494 41 2 v 1553 2523 a Fl(\015)1573 2513 y Fe(h)1598 2457 y Fk(\021)1623 2469 y Fl(\013)1669 2505 y Fp(with)f Fn(\013)i Fp(=)907 2728 y(26)p eop %%Page: 27 27 27 26 bop -13 -5 a Fp(0)p Fn(;)8 b Fp(1)p Fn(;)g Fp(2)17 b(\(see)f(lemm)o(a)e(2.1\),)i(a)h(sligh)o(t)f(mo)q(di\014cation)g(of)g (the)g(pro)q(of)i(in)e(\(B.G.P)l(.S.\)\(B.M.)o(\),)d(essen)o(tially)-13 74 y(based)k(on)f(the)g(Grahm-Hadamard)g(inequalit)o(y)e(\(Le.\),)h (sho)o(ws)j(that)e(if)294 223 y(max)295 253 y Fl(i;k)q Fi(\025)p Fl(h)393 223 y Fm(j)r Fp(\026)-26 b Fn(v)431 230 y Fl(i;k)474 223 y Fm(j)p Fn(;)8 b Fm(j)p Fn(\027)548 230 y Fl(k)569 223 y Fm(j)14 b(\024)i Fp(\026)-27 b Fn(")115 b Fp(max)783 247 y Fe(i)p Ff(=0)p Fe(;)p Ff(1)p Fe(;)p Ff(2)803 275 y Fe(k)q Fh(\025)p Fe(h)895 149 y Fk(\014)895 174 y(\014)895 199 y(\014)895 223 y(\014)895 248 y(\014)914 190 y Fn(\033)942 197 y Fl(k)p 914 212 50 2 v 914 258 a Fn(\015)942 243 y Fl(k)968 149 y Fk(\014)968 174 y(\014)968 199 y(\014)968 223 y(\014)968 248 y(\014)982 162 y Fl(i)1010 223 y Fm(\024)1074 211 y Fp(~)1062 223 y Fn(C)110 b Fp(max)1219 253 y Fl(k)q Fi(\025)p Fl(h)1333 190 y Fn(Z)1366 197 y Fl(k)p 1311 212 100 2 v 1311 258 a Fn(Z)1344 265 y Fl(k)q Fi(\000)p Fj(1)1429 223 y 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Fm(\024)15 b Fn(K)20 b Fp(for)d(an)o(y)g Fn(k)g(>)d(h)j Fp(then)g(max)1056 81 y Fl(i;k)q Fi(\025)p Fl(h)1156 74 y Fm(j)r Fp(\026)-26 b Fn(v)1194 81 y Fl(i;k)1236 74 y Fm(j)15 b(\024)i Fp(^)-27 b Fn(")17 b Fp(and)g(max)1544 81 y Fl(k)q Fi(\025)p Fl(h)1646 53 y(Z)1670 59 y Fe(k)p 1626 62 83 2 v 1626 91 a Fl(Z)1650 97 y Fe(k)q Fh(\000)p Ff(1)1728 74 y Fm(\024)d Fn(e)1804 55 y Fl(\014)1824 60 y Ff(1)1843 55 y Fj(^)-20 b Fl(")1857 44 y Ff(2)-13 152 y Fp(with)19 b(^)-27 b Fn(")14 b Fm(\024)i Fp(\026)-27 b Fn(")p Fp(.)21 b(This)c(follo)o(ws)f(from)f(the)h(ab)q(o) o(v)o(e)g(lemma,)d(noting)k(that)f(for)h Fm(j)p Fn(\025)p Fm(j)f Fp(small)f(enough:)700 292 y Fn(\025)736 238 y Fl(h)p Fj(+1)739 250 y Fk(X)737 343 y Fl(k)q Fj(=0)816 258 y Fn(\033)844 265 y Fl(k)p 815 280 50 2 v 815 326 a Fn(\015)843 312 y Fl(k)884 292 y Fm(\024)e Fn(\025)976 279 y Fp(^)964 292 y Fn(C)1008 258 y(\033)1036 265 y Fl(h)p 1008 280 51 2 v 1008 326 a Fn(\015)1036 312 y Fl(h)1077 292 y Fn(<)1138 258 y Fp(\026)-27 b Fn(")p 1134 280 25 2 v 1134 326 a Fp(2)1790 292 y(\(85\))-13 444 y(for)17 b(a)h(suitable)f(constan)o(t)495 431 y(^)484 444 y Fn(C)t Fp(,)g(as)h(from)e(eq.\(83\))901 419 y Fl(\033)921 425 y Fe(h)p Fh(\000)p Ff(1)p 901 432 80 2 v 901 461 a Fl(\015)921 451 y Fe(h)p Fh(\000)p Ff(1)990 422 y Fl(\015)1010 410 y Fe(h)p 990 432 41 2 v 990 461 a Fl(\033)1010 467 y Fe(h)1051 444 y Fm(\025)f Fn(C)k(>)c Fp(1.)25 b(In)17 b(other)g(w)o(ords,)h(from)e(lemm)o(a)-13 522 y(2.2)g(w)o(e)g(kno)o(w)g (that)h(it)f(is)g(p)q(ossible)g(to)g(c)o(ho)q(ose)h Fn(\025)g Fp(so)f(that)h(the)f(Beta)g(function)g Fn(\014)1497 504 y Fl(k)1494 534 y(i)1534 522 y Fp(is)f(analytic)h(in)g(its)-13 600 y(argumen)o(ts)h(for)i Fn(k)h Fm(\025)d Fn(h)p Fp(.)29 b(W)l(e)18 b(can)g(then)h(c)o(ho)q(ose)g Fn(\025)g Fp(so)g(that)g Fn(\014)1171 582 y Fl(k)1168 612 y(i)1210 600 y Fp(is)f(analytic)g(in)g (its)h(argumen)o(ts)e(for)-13 678 y Fn(k)f Fm(\025)d Fn(h)108 660 y Fi(\003)145 678 y Fp(where)j Fn(h)314 660 y Fi(\003)350 678 y Fp(is)g(de\014ned)g(so)h(that)738 657 y Fl(\033)758 663 y Fe(h)p 738 667 V 738 695 a Fl(\015)758 686 y Fe(h)797 678 y Fm(\024)d Fp(1)i(for)h(an)o(y)f Fn(h)e(>)g(h)1179 660 y Fi(\003)1215 678 y Fp(but)1310 656 y Fl(\033)1330 663 y Fe(h)1348 656 y Fh(\003)p 1310 667 59 2 v 1310 696 a Fl(\015)1330 687 y Fe(h)1348 680 y Fh(\003)1387 678 y Fn(>)g Fp(1;)i(from)f(lemma)e(2.2)k(it)-13 757 y(is)f(easy)g(to)h(see)f(that:)633 805 y(log)696 817 y Fl(\015)727 805 y Fn(u)10 b Fp(+)h(1)p 633 830 206 2 v 635 876 a(1)h(+)f Fn(\025\014)776 883 y Fj(3)796 876 y Fn(c)817 883 y Fj(3)858 841 y Fm(\024)i Fn(h)938 821 y Fi(\003)972 841 y Fm(\024)1069 805 y Fp(log)1132 817 y Fl(\015)1163 805 y Fn(u)p 1029 830 201 2 v 1029 876 a Fp(1)f(+)f Fn(\025\014)1170 883 y Fj(3)1190 876 y Fn(c)1211 883 y Fj(4)1790 841 y Fp(\(86\))-13 948 y(The)19 b(in)o(tro)q(duction)f(of)h(the)g(scale)f Fn(h)663 930 y Fi(\003)702 948 y Fp(is)h(crucial)e(and)j(represen)o(ts)e(the)h(ma)s (jor)e(di\013erence)h(with)h(the)-13 1026 y(not)c(\014lled)f(band)h (case,)g(see)f(\(B.M.\);)f(the)h("in)o(tuitiv)o(e")f(meaning)h(of)h(it) f(will)g(b)q(e)h(discussed)f(at)h(the)g(end)-13 1104 y(of)h(this)g(section.)10 1183 y(F)l(rom)h(lemm)o(a)e(2.2)j(and)h(the)e (ab)q(o)o(v)o(e)h(considerations)g(it)f(follo)o(ws)g(that)h Fn(V)1377 1164 y Fl(h)1397 1153 y Fh(\003)1417 1183 y Fp(\()1436 1144 y Fm(p)p 1478 1144 74 2 v 39 x Fn(Z)1511 1190 y Fl(h)1531 1181 y Fh(\003)1551 1183 y Fn( )1585 1164 y Fi(\024)p Fl(h)1632 1153 y Fh(\003)1653 1183 y Fp(\))f(is)h(an)g(an-)-13 1261 y(alytic)f(function)i(of)g Fn(\025)g Fp(for)g Fm(j)p Fn(\025)p Fm(j)g Fp(small)e(enough;)j(in)e (order)h(to)g(compute)e(the)h(partition)h(function)37 b(it)-13 1339 y(remains)15 b(to)h(p)q(erform)f(the)i(in)o(tegration:) 545 1460 y Fm(N)k Fp(=)659 1401 y Fk(Z)709 1460 y Fn(P)740 1467 y Fl(Z)764 1474 y Fe(h)782 1467 y Fh(\003)804 1460 y Fp(\()p Fn(d )882 1439 y Fi(\024)p Fl(h)929 1427 y Fh(\003)950 1460 y Fp(\))p Fn(e)992 1439 y Fi(\000)p Fl(V)1047 1427 y Fe(h)1065 1417 y Fh(\003)1085 1439 y Fj(\()1099 1406 y Fm(p)p 1141 1406 63 2 v 33 x Fl(Z)1165 1446 y Fe(h)1183 1439 y Fh(\003)t Fj(\))p Fl( )1241 1427 y Fh(\024)p Fe(h)1283 1417 y Fh(\003)1303 1439 y Fj(\))1790 1460 y Fp(\(87\))-13 1580 y(The)12 b(propagator)j(asso)q(ciated)e(with) f(suc)o(h)g(in)o(tegration,)h(written)e(in)h(term)f(of)i(particle)e (\014elds,)h(is)g(giv)o(en)-13 1658 y(b)o(y)l(,)j(from)g(eq.\(67\):)241 1779 y Fn(g)266 1758 y Fi(\024)p Fl(h)313 1746 y Fh(\003)334 1779 y Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)517 1737 y Fk(X)513 1830 y Fl(~)-18 b(!)q(;~)g(!)569 1821 y Fh(0)589 1720 y Fk(Z)638 1779 y Fn(dk)r(e)713 1758 y Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))840 1779 y Fn(C)879 1758 y Fi(\000)p Fj(1)875 1791 y Fl(h)895 1782 y Fh(\003)926 1779 y Fp(\()p Fn(k)r Fp(\))p Fn(T)1027 1758 y Fi(\000)p Fj(1)1020 1791 y Fl(h)1040 1782 y Fh(\003)1073 1779 y Fp(\()p Fn(k)r Fp(\))1138 1786 y Fl(~)g(!)q(;~)g(!)1194 1777 y Fh(0)1208 1779 y Fn(e)1231 1758 y Fi(\000)p Fl(i)p Fj(\()p Fl(!)q(x)p Fi(\000)p Fl(!)1377 1746 y Fh(0)1388 1758 y Fl(y)q Fj(\))1436 1779 y Fp(=)241 1934 y(=)293 1876 y Fk(Z)343 1934 y Fn(dk)458 1900 y Fp(1)p 400 1922 139 2 v 400 1968 a Fn(Z)433 1975 y Fl(h)453 1966 y Fh(\003)474 1968 y Fp(\()p Fn(k)r Fp(\))726 1900 y Fn(C)765 1880 y Fi(\000)p Fj(1)761 1913 y Fl(h)781 1904 y Fh(\003)812 1900 y Fp(\()p Fn(k)r Fp(\))p Fn(e)900 1882 y Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))p 549 1922 654 2 v 549 1976 a Fp(\()p Fm(\000)p Fn(ik)649 1983 y Fj(0)680 1976 y Fp(+)726 1963 y Fn(~)729 1976 y(k)756 1962 y Fj(2)775 1976 y Fp(\))794 1962 y Fj(2)825 1976 y Fm(\000)11 b Fp(\()p Fn(\031)921 1963 y(~)924 1976 y(k)q Fp(\))969 1962 y Fj(2)1000 1976 y Fm(\000)g Fn(\033)1078 1983 y Fl(h)1098 1974 y Fh(\003)1118 1976 y Fp(\()p Fn(k)r Fp(\))1183 1962 y Fj(2)1790 1934 y Fp(\(88\))241 2054 y([cos\()p Fn(\031)r Fp(\()p Fn(x)f Fm(\000)h Fn(y)r Fp(\)\)\()p Fm(\000)p Fn(ik)640 2061 y Fj(0)670 2054 y Fp(+)717 2041 y Fn(~)719 2054 y(k)746 2033 y Fj(2)766 2054 y Fp(\))g(+)g Fn(i\031)889 2041 y(~)892 2054 y(k)f Fp(sin)o(\()p Fn(\031)r Fp(\()p Fn(x)h Fm(\000)f Fn(y)r Fp(\)\))h(+)g Fn(\033)1294 2061 y Fl(h)1314 2052 y Fh(\003)1334 2054 y Fp(\()p Fn(k)r Fp(\))d(cos)q(\()p Fn(\031)r Fp(\()p Fn(x)i Fp(+)h Fn(y)r Fp(\)\)])-13 2174 y(from)k(a)i(standard)g(in)o(tegration)f(b)o(y)g (parts)h(it)e(follo)o(ws)i(that:)547 2331 y Fm(j)p Fn(g)586 2310 y Fi(\024)p Fl(h)633 2299 y Fh(\003)654 2331 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fm(j)13 b(\024)1013 2297 y Fn(C)1048 2304 y Fl(N)1082 2297 y Fn(\015)1110 2279 y Fl(h)1130 2267 y Fh(\003)p 852 2319 460 2 v 852 2365 a Fn(Z)885 2372 y Fl(h)905 2363 y Fh(\003)926 2365 y Fp(\(1)e(+)g(\()p Fm(j)p Fn(x)g Fm(\000)g Fn(y)r Fm(j)p Fn(\015)1219 2351 y Fl(h)1239 2341 y Fh(\003)1259 2365 y Fp(\))1278 2351 y Fl(N)-13 2446 y Fg(i.e.)17 b Fp(it)f(ob)q(eys)h(to)g(the)f(same)f(b)q (ound)i(as)g Fn(g)757 2428 y Fl(h)755 2459 y(~)-18 b(!)777 2464 y Ff(1)795 2459 y Fl(;~)g(!)827 2464 y Ff(2)846 2446 y Fp(\()p Fn(x)11 b Fm(\000)f Fn(y)r Fp(\))16 b(for)h Fn(h)d(>)f(h)1210 2428 y Fi(\003)1230 2446 y Fp(,)j(see)g(lemm)o(a)e (2.1.)10 2525 y(This)k(follo)o(ws)f(noting)g(that)h(\()p Fm(\000)p Fn(ik)643 2532 y Fj(0)674 2525 y Fp(+)721 2512 y Fn(~)723 2525 y(k)750 2507 y Fj(2)770 2525 y Fp(\))789 2507 y Fj(2)820 2525 y Fm(\000)12 b Fp(\()p Fn(\031)917 2512 y(~)920 2525 y(k)q Fp(\))965 2507 y Fj(2)996 2525 y Fm(\000)g Fn(\033)1075 2532 y Fl(h)1095 2523 y Fh(\003)1115 2525 y Fp(\()p Fn(k)r Fp(\))1180 2507 y Fj(2)1215 2525 y Fp(=)j Fn(\015)1296 2507 y Fj(2)p Fl(h)1334 2495 y Fh(\003)1354 2525 y Fn(D)q Fp(\()p Fn(\015)1442 2507 y Fl(h)1462 2495 y Fh(\003)1483 2525 y Fn(k)r Fp(\))i(and)h(eac)o(h)e (of)i(the)-13 2603 y(tree)13 b(term)e(in)j(the)f(n)o(umerator)f(can)i (b)q(e)g(written)f(as)h Fn(\015)969 2585 y Fl(h)989 2573 y Fh(\003)1009 2603 y Fn(N)5 b Fp(\()p Fn(\015)1100 2585 y Fi(\000)p Fl(h)1147 2573 y Fh(\003)1169 2603 y Fn(k)r Fp(\))13 b(with)g Fn(D)q Fp(\()p Fn(k)r Fp(\))p Fn(;)23 b(N)5 b Fp(\()p Fn(k)r Fp(\))14 b(b)q(ounded)g(and)907 2728 y(30)p eop %%Page: 31 31 31 30 bop -13 -5 a Fn(O)q Fp(\(1\))18 b(with)f(all)g(their)f(deriv)m (ativ)o(es.)23 b(Then)17 b(also)h(the)f(in)o(tegration)g(eq.\(87\))g (is)g(w)o(ell)f(de\014ned)h(and)h(the)-13 74 y(analysis)e(of)h(the)f (partition)g(function)g Fm(N)24 b Fp(is)16 b(complete.)10 211 y(The)c(Sc)o(h)o(winger)f(function)g(eq.\(26\))h(admits)e(a)i(tree) f(expansion)h(similar)e(to)i(the)f(one)h(of)g(the)g(partition)-13 289 y(function,)19 b(whose)g(con)o(v)o(ergence)f(follo)o(ws)g(from)g (the)h(partition)g(function)f(expansion)i(con)o(v)o(ergence,)-13 367 y(see)d(\(B.G.P)l(.S.\);)f(from)g(brevit)o(y)g(w)o(e)h(do)h(not)g (rep)q(ort)g(it)f(here)g(but)g(w)o(e)g(simply)f(quote)h(the)g(results)g (of)-13 446 y(the)f(functional)g(in)o(tegration)g(in)g(eq.\(26\):)247 583 y Fn(S)s Fp(\()p Fn(x;)8 b(y)r Fp(\))40 b(=)514 525 y Fk(Z)563 583 y Fn(dk)620 550 y(e)643 532 y Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))p 621 572 150 2 v 640 618 a Fp(\(2)p Fn(\031)r Fp(\))732 603 y Fj(2)780 550 y Fn(h)p Fp(\()p Fn(k)854 532 y Fj(2)852 562 y(0)885 550 y Fp(+)11 b Fn(E)s Fp(\()989 537 y Fn(~)992 550 y(k)r Fp(\))1038 532 y Fj(2)1058 550 y Fp(\))p 780 572 297 2 v 796 625 a Fm(\000)p Fn(ik)877 632 y Fj(0)908 625 y Fp(+)g Fn(E)s Fp(\()1012 612 y Fn(~)1015 625 y(k)r Fp(\))1093 583 y(+)434 734 y(+)570 680 y Fj(0)549 693 y Fk(X)514 785 y Fl(h)p Fj(=)p Fl(h)581 775 y Fh(\003)600 785 y Fi(\000)p Fj(1)667 693 y Fk(X)653 786 y Fl(~)-18 b(!)675 791 y Ff(1)692 786 y Fl(;~)g(!)724 791 y Ff(2)750 734 y Fn(e)773 713 y Fl(i)790 700 y Fe(\031)p 789 706 19 2 v 790 727 a(a)813 713 y Fj(\()p Fl(~)g(!)849 718 y Ff(1)866 713 y Fl(~)g(x)p Fi(\000)p Fl(~)g(!)935 718 y Ff(2)953 713 y Fl(~)g(y)q Fj(\))987 734 y Fp(\()p Fn(g)1031 713 y Fl(h)1029 746 y(~)g(!)1051 751 y Ff(1)1069 746 y Fl(;~)g(!)1101 751 y Ff(2)1120 734 y Fp(\()p Fn(x)11 b Fm(\000)f Fn(y)r Fp(\))h(+)i(\026)-26 b Fn(g)1357 713 y Fl(h)1355 746 y(~)-18 b(!)1377 751 y Ff(1)1395 746 y Fl(;~)g(!)1427 751 y Ff(2)1446 734 y Fp(\()p Fn(x)10 b Fm(\000)h Fn(y)r Fp(\)\))173 b(\(89\))-13 882 y(where)16 b(w)o(e)g(call)g Fn(g)315 864 y Fl()f(\015)1648 2254 y Fl(h)1668 2242 y Fh(\003)1707 2272 y Fp(but)i(still)109 2350 y Fn(<<)13 b Fp(1)k(w)o(e)f(ha)o(v)o(e)660 2475 y Fn(")p Fp(\()699 2462 y Fn(~)702 2475 y(k)d Fp(+)e Fn(~)-24 b(!)r(p)845 2482 y Fl(F)875 2475 y Fn(;)8 b(u)p Fp(\))i Fm(\000)h Fn(\026)j Fm(')704 b Fp(\(97\))660 2578 y(sign\()760 2564 y Fn(~)763 2578 y(k)r Fp(\))809 2515 y Fk(q)p 851 2515 155 2 v 848 2564 a Fn(~)851 2578 y(k)878 2563 y Fj(2)908 2578 y Fp(+)11 b Fn(u)985 2563 y Fj(2)1019 2578 y Fm(')i Fp(2)p Fn(\031)1122 2564 y(~)1125 2578 y(k)r Fp(\(1)f(+)f Fn(O)q Fp(\()1318 2544 y Fn(u)1346 2526 y Fj(2)p 1318 2566 48 2 v 1316 2606 a Fn(~)1318 2620 y(k)1345 2605 y Fj(2)1371 2578 y Fp(\)\))907 2728 y(32)p eop %%Page: 33 33 33 32 bop 109 -5 a Fp(while)15 b(in)h(the)g(region)g Fm(j)534 -18 y Fn(~)537 -5 y(k)r Fm(j)e Fn(<)g(\015)672 -23 y Fl(h)692 -34 y Fh(\003)728 -5 y Fp(w)o(e)i(ha)o(v)o(e:)631 134 y Fn(")p Fp(\()670 121 y Fn(~)673 134 y(k)c Fp(+)f Fn(~)-24 b(!)s(p)816 141 y Fl(F)845 134 y Fn(;)8 b(u)p Fp(\))j Fm(\000)g Fn(\026)j Fm(')g Fn(u)p Fp(\(1)d(+)g Fn(O)q Fp(\()1262 87 y Fn(~)1264 101 y(k)1291 82 y Fj(2)p 1264 123 48 2 v 1264 168 a Fn(u)1292 154 y Fj(2)1317 134 y Fp(\)\))435 b(\(98\))109 252 y(The)15 b(meaning)f(of)h(the)g (scale)g Fn(h)683 234 y Fi(\003)718 252 y Fp(is)g(then)g(clear:)20 b(it)14 b(separates)i(the)f(momen)o(ta)e(near)i(the)g(F)l(ermi)109 331 y(surface)i(in)f(t)o(w)o(o)h(regions,)g(one)g(in)f(whic)o(h)h(the)f (disp)q(ersion)h(relation)g(is)g(appro)o(ximately)d(linear)109 409 y(in)165 396 y Fn(~)167 409 y(k)20 b Fp(and)e(another)h(in)e(whic)o (h)g(is)h(quadratic.)26 b(The)17 b(t)o(w)o(o)h(p)q(oin)o(ts)g(Sc)o(h)o (winger)f(function)h(has)g(of)109 487 y(course)f(a)h(di\013eren)o(t)e (b)q(eha)o(vior)h(for)h(small)896 474 y Fn(~)898 487 y(k)i Fp(in)d(these)g(t)o(w)o(o)g(regions.)24 b(In)17 b(the)g Fn(\025)f Fm(6)p Fp(=)g Fn(\027)i Fm(6)p Fp(=)e(0)i(case)109 565 y Fn(\033)137 572 y Fl(h)179 565 y Fp(is)i(not)h(constan)o(t)f(and) h(this)f(has)h(the)f(e\013ect)g(that)g Fn(h)1170 547 y Fi(\003)1210 565 y Fp(is)g(di\013eren)o(t)f(with)h(resp)q(ect)g(to)h (the)109 644 y Fn(\025)g Fp(=)g(0)g(case)f(but)h(its)f(meaning)f (remains)g(the)i(same)e Fg(i.e.)f Fp(it)i(separates)h(the)f(momen)o(ta) e(in)o(to)109 722 y(t)o(w)o(o)c(regions)g(in)f(whic)o(h)h(the)f(t)o(w)o (o)h(p)q(oin)o(ts)h(Sc)o(h)o(winger)e(function)g(has)i(a)f(di\013eren)o (t)g(b)q(eha)o(viour.)20 b(If)109 800 y Fn(u)14 b Fp(=)g(0)p Fn(;)8 b(\025)14 b Fm(6)p Fp(=)g(0)j(one)g(\014nds)g(\(see)f(\(B.G.P)l (.S.\)\))e(a)j(form)o(ula)e(similar)f(to)j(eq.\(89\))f(with)h Fn(h)1689 782 y Fi(\003)1722 800 y Fp(=)d Fm(\0001)p Fp(,)109 878 y Fn(\033)137 885 y Fl(h)157 876 y Fh(\003)194 878 y Fp(=)i(0)i(and)h Fn(Z)420 885 y Fl(h)459 878 y Fm(')e Fn(\015)543 860 y Fi(\000)p Fl(\021)588 865 y Ff(1)605 860 y Fl(h)646 878 y Fg(i.e.)g Fp(the)h(p)q(erio)q(dic)g(p)q (oten)o(tial)f(has)i(the)f(e\013ect)f(that)h(the)g(infrared)109 957 y(scales)f(are)g(\014nite.)22 b(It)17 b(is)g(clear)f(that)i(then)e (the)h(scale)g Fn(h)1143 939 y Fi(\003)1180 957 y Fp(separates)g(a)h (region)f(in)f(whic)o(h)h Fn(S)s Fp(\()p Fn(k)r Fp(\))109 1035 y(b)q(eha)o(v)o(es)d(lik)o(e)f(the)i(Sc)o(h)o(winger)f(function)g (of)h(the)g Fn(u)f Fp(=)f(0)j(case)f Fg(i.e.)i Fp(as)f Fm(j)p Fn(k)r Fm(j)1448 1017 y Fi(\000)p Fj(\(1)p Fi(\000)p Fl(\021)1552 1022 y Ff(1)1569 1017 y Fj(\))1584 1035 y Fn(S)1614 1042 y Fj(0)1649 1035 y Fp(to)f(a)h(region)109 1113 y(in)g(whic)o(h)f(the)h(b)q(eha)o(viour)g(is)h(giv)o(en)e(b)o(y)h (eq\(18\).)-13 1230 y Fd(Remark)g(2)25 b Fp(:)20 b(Of)c(course)f(the)g (scale)g Fn(h)730 1212 y Fi(\003)766 1230 y Fp(should)h(b)q(e)f(c)o (hosen)g(so)h(that)1309 1209 y Fl(\033)1329 1215 y Fe(h)p 1309 1218 41 2 v 1309 1247 a Fl(\015)1329 1238 y Fe(h)1368 1230 y Fm(\024)e Fn(K)19 b Fp(for)d(an)o(y)f Fn(h)f(>)g(h)1768 1212 y Fi(\003)1803 1230 y Fp(but)114 1286 y Fl(\033)134 1293 y Fe(h)152 1286 y Fh(\003)p 114 1297 59 2 v 114 1327 a Fl(\015)134 1317 y Fe(h)152 1310 y Fh(\003)195 1308 y Fn(>)19 b(K)k Fp(with)18 b(an)o(y)h(constan)o(t)h Fn(K)j Fp(\(w)o(e)18 b(c)o(hose)h Fn(K)j Fp(=)c(1\).)30 b(In)18 b(this)h(case)g(it)f(is)h(easy)g(to)g(c)o(hec)o(k)109 1387 y(that)14 b Fm(j)p Fn(\025)p Fm(j)g(\024)f Fp(min)o([)p Fn(K)q(;)506 1367 y Fj(1)p 498 1375 33 2 v 498 1404 a Fl(K)535 1387 y Fp(])h(and)g Fn(\033)683 1394 y Fl(h)703 1385 y Fh(\003)737 1387 y Fm(')f Fn(u)817 1369 y Fj(1)p Fi(\000)p Fl(\021)880 1374 y Ff(2)899 1387 y Fn(K)944 1369 y Fl(\021)962 1374 y Ff(2)981 1387 y Fp(,)h Fn(Z)1042 1394 y Fl(h)1062 1385 y Fh(\003)1097 1387 y Fm(')f Fn(u)1177 1369 y Fl(\021)1195 1374 y Ff(1)1214 1387 y Fn(K)1259 1369 y Fi(\000)p Fl(\021)1304 1374 y Ff(1)1337 1387 y Fp(so)i(that)f(eq.\(17\))f(still)g(holds.)109 1465 y(W)l(e)19 b(ha)o(v)o(e)g(then)g(some)g(freedom)f(in)h(the)h(c)o(hoice)e(of)i Fn(h)1124 1447 y Fi(\003)1163 1465 y Fp(but)g(of)g(course)g(not)g(to)q (o)g(m)o(uc)o(h.)29 b(F)l(or)109 1543 y(instance)16 b(if)h(w)o(e)f(c)o (ho)q(ose)i Fn(h)601 1525 y Fi(\003)636 1543 y Fp(=)d(log)752 1555 y Fl(\015)782 1543 y Fn(u)i Fp(then)g Fm(j)p Fn(g)978 1525 y Fi(\024)p Fl(h)1025 1513 y Fh(\003)1045 1543 y Fp(\()p Fn(x;)8 b(y)r Fp(\))p Fm(j)14 b(\024)1373 1522 y Fl(C)1398 1528 y Fe(N)1427 1522 y Fl(\015)1447 1510 y Ff(2)p Fe(h)1480 1500 y Fh(\003)p 1246 1531 381 2 v 1246 1561 a Fl(Z)1270 1568 y Fe(h)1288 1561 y Fh(\003)t Fl(\033)1328 1568 y Fe(h)1346 1561 y Fh(\003)5 b Fj(\(1+\()p Fi(j)p Fl(x)p Fi(\000)p Fl(y)q Fi(j)p Fl(\033)1546 1552 y Fe(h)1564 1545 y Fh(\003)1584 1561 y Fj(\))1598 1552 y Fe(N)1649 1543 y Fg(i.e.)17 b Fp(it)g(do)q(es)109 1621 y(not)f(ob)q(ey)h(to)g(the)f(same)f(b)q(ound)i(of)g Fn(g)809 1603 y Fl(h)831 1621 y Fp(\()p Fn(x)11 b Fm(\000)g Fn(y)r Fp(\))16 b(for)g Fn(h)e Fm(\025)g Fn(h)1197 1603 y Fi(\003)1216 1621 y Fp(.)-13 1787 y Fc(2.4)69 b(Outlo)r(ok)23 b(on)g(the)g(spinning) f(case)-13 1897 y Fp(Let)c(us)h(discuss)g(\014nally)e(what)i(happ)q (ens)h(in)e(the)g(spinning)g(case.)28 b(A)18 b(second)h(order)f (computations)-13 1975 y(sho)o(ws)f(that:)498 2094 y Fn(g)521 2101 y Fj(1)p Fl(;h)p Fi(\000)p Fj(1)630 2094 y Fp(=)d Fn(g)705 2101 y Fj(1)p Fl(;h)766 2094 y Fm(\000)d Fp(\()p Fn(\014)i Fp(+)e Fn(O)q Fp(\()987 2060 y Fn(\033)1015 2067 y Fl(h)p 987 2082 51 2 v 987 2128 a Fn(\015)1015 2113 y Fl(h)1043 2094 y Fp(\)\))p Fn(g)1106 2073 y Fj(2)1104 2106 y(1)p Fl(;h)498 2209 y Fn(g)521 2216 y Fj(2)p Fl(;h)p Fi(\000)p Fj(1)630 2209 y Fp(=)j Fn(g)705 2216 y Fj(2)p Fl(;h)766 2209 y Fp(+)d(\()p Fn(\014)i Fp(+)e Fn(O)q Fp(\()986 2176 y Fn(\033)1014 2183 y Fl(h)p 986 2198 V 986 2243 a Fn(\015)1014 2229 y Fl(h)1042 2209 y Fp(\)\))p Fn(g)1105 2189 y Fj(2)1103 2222 y(1)p Fl(;h)498 2325 y Fn(g)521 2332 y Fj(3)p Fl(;h)p Fi(\000)p Fj(1)630 2325 y Fp(=)j Fn(g)705 2332 y Fj(3)p Fl(;h)766 2325 y Fm(\000)d Fp(2\()p Fn(\014)j Fp(+)d Fn(O)q Fp(\()1023 2291 y Fn(\033)p 1012 2313 V 1012 2359 a(\015)1040 2345 y Fl(h)1068 2325 y Fp(\)\))p Fn(g)1129 2332 y Fj(3)p Fl(;h)1179 2325 y Fp(\()p Fn(g)1221 2332 y Fj(1)p Fl(;h)1282 2325 y Fm(\000)g Fp(2)p Fn(g)1379 2332 y Fj(2)p Fl(;h)1429 2325 y Fp(\))498 2440 y Fn(g)521 2447 y Fj(4)p Fl(;h)p Fi(\000)p Fj(1)630 2440 y Fp(=)j Fn(g)705 2447 y Fj(4)p Fl(;h)766 2440 y Fp(+)d Fn(O)q Fp(\()877 2407 y Fn(\033)905 2414 y Fl(h)p 877 2429 V 877 2475 a Fn(\015)905 2460 y Fl(h)933 2440 y Fp(\))498 2576 y Fn(g)521 2583 y Fj(5)p Fl(;h)p Fi(\000)p Fj(1)630 2576 y Fp(=)j Fn(g)705 2583 y Fj(5)p Fl(;h)766 2576 y Fm(\000)d Fp(\()p Fn(\014)i Fp(+)e Fn(O)q Fp(\()987 2542 y Fn(\033)1017 2524 y Fl(h)p 987 2564 52 2 v 988 2610 a Fn(\015)1016 2595 y Fl(h)1045 2576 y Fp(\)\))p Fn(g)1106 2583 y Fj(5)p Fl(;h)1156 2576 y Fp(\(2)p Fn(g)1222 2583 y Fj(1)p Fl(;h)1284 2576 y Fm(\000)f Fn(g)1356 2583 y Fj(2)p Fl(;h)1407 2576 y Fp(\))907 2728 y(33)p eop %%Page: 34 34 34 33 bop 498 20 a Fn(g)521 27 y Fj(6)p Fl(;h)p Fi(\000)p Fj(1)630 20 y Fp(=)14 b Fn(g)705 27 y Fj(6)p Fl(;h)766 20 y Fm(\000)d Fp(\()p Fn(\014)i Fp(+)e Fn(O)q Fp(\()987 -13 y Fn(\033)1017 -31 y Fl(h)p 987 9 52 2 v 988 54 a Fn(\015)1016 40 y Fl(h)1045 20 y Fp(\)\))p Fn(g)1106 27 y Fj(6)p Fl(;h)1156 20 y Fp(\(2)p Fn(g)1222 27 y Fj(1)p Fl(;h)1284 20 y Fm(\000)f Fn(g)1356 27 y Fj(2)p Fl(;h)1407 20 y Fp(\))498 156 y Fn(\033)526 163 y Fl(h)p Fi(\000)p Fj(1)607 156 y Fp(=)k Fn(\033)687 163 y Fl(h)720 156 y Fp(+)d(\()p Fn(\014)816 163 y Fj(3)846 156 y Fp(+)g Fn(O)q Fp(\()957 122 y Fn(\033)987 104 y Fl(h)p 957 144 V 958 190 a Fn(\015)986 175 y Fl(h)1015 156 y Fp(\)\))p Fn(g)1076 163 y Fj(2)p Fl(;h)498 262 y Fn(\034)519 269 y Fl(h)p Fi(\000)p Fj(1)600 262 y Fp(=)j Fn(\034)673 269 y Fl(h)707 262 y Fp(+)763 249 y(~)756 262 y Fn(\014)r(g)809 269 y Fj(2)p Fl(;h)859 262 y Fn(g)882 269 y Fj(5)p Fl(;h)498 353 y Fn(\022)521 360 y Fl(h)p Fi(\000)p Fj(1)602 353 y Fp(=)g Fn(\022)677 360 y Fl(h)710 353 y Fp(+)766 340 y(~)759 353 y Fn(\014)r(g)812 360 y Fj(2)p Fl(;h)863 353 y Fn(g)886 360 y Fj(5)p Fl(;h)498 444 y Fn(\016)520 451 y Fl(h)p Fi(\000)p Fj(1)601 444 y Fp(=)g Fn(\016)675 451 y Fl(h)708 444 y Fp(+)d Fn(O)q Fp(\()819 410 y Fn(\033)847 417 y Fl(h)p 819 432 51 2 v 819 478 a Fn(\015)847 463 y Fl(h)875 444 y Fp(\))503 538 y Fn(Z)536 545 y Fl(h)p Fi(\000)p Fj(1)p 503 560 101 2 v 525 606 a Fn(Z)558 613 y Fl(h)623 572 y Fp(=)i(1)f(+)f Fn(\014)787 579 y Fj(1)806 572 y Fn(g)831 551 y Fj(2)829 584 y(2)p Fl(;h)890 572 y Fp(+)g Fn(O)q Fp(\()1001 538 y Fn(\033)1029 545 y Fl(h)p 1001 560 51 2 v 1001 606 a Fn(\015)1029 592 y Fl(h)1057 572 y Fp(\))714 b(\(99\))-13 704 y(with)15 b Fn(\014)s(;)8 b(\014)178 711 y Fj(1)197 704 y Fn(;)g(\014)247 711 y Fj(3)280 704 y Fn(>)14 b Fp(0.)21 b(One)15 b(could)h(pro)q(ceed)g(as)g (in)f(the)h(spinless)f(case)h(using)g(the)g(anomalous)g(scaling)-13 783 y(of)21 b(sec.\(2.2\).)34 b(Ho)o(w)o(ev)o(er)19 b(in)i(the)f (spinning)h(case)g(the)g(running)g(coupling)f(constan)o(ts)i(do)f(not)h (sta)o(y)-13 861 y(necessarily)15 b(b)q(ounded)i(un)o(til)f(the)g (scale)g Fn(h)774 843 y Fi(\003)810 861 y Fp(de\014ned)g(as)h(in)f (eq.\(86\).)21 b(In)16 b(fact)g(b)o(y)g(eq.\(99\),)g(assuming)-13 939 y(heuristically)i(that)i(the)g(b)q(eta)g(function)g(higher)g (orders)g(v)m(anish,)h(it)e(follo)o(ws)h(that)h Fn(g)1592 946 y Fj(1)p Fl(;h)1662 939 y Fm(')1769 915 y Fl(g)1786 920 y Ff(1)p Fe(;)p Ff(0)p 1726 927 146 2 v 1726 956 a Fj(1)p Fi(\000)p Fl(\014)r(g)1810 961 y Ff(1)p Fe(;)p Ff(0)1851 956 y Fl(h)-13 1017 y Fp(whic)o(h)e(is)h(b)q(ounded)h(for)g Fn(h)f Fm(\025)g Fn(h)600 999 y Fi(\003)640 1017 y Fp(for)g(an)o(y)g Fn(u)g Fg(only)i(if)e Fn(g)1045 1024 y Fj(1)p Fl(;)p Fj(0)1113 1017 y Fn(>)g Fp(0;)i(if)e Fn(g)1303 1024 y Fj(1)p Fl(;)p Fj(0)1371 1017 y Fn(<)g Fp(0)g(w)o(e)g(ha)o(v)o(e)f(to)i (require)-13 1096 y(that)c Fm(j)p Fn(h)135 1077 y Fi(\003)155 1096 y Fm(j)e Fn(<)g(O)q Fp(\()320 1076 y Fj(1)p 299 1084 59 2 v 299 1113 a Fl(g)316 1118 y Ff(1)p Fe(;)p Ff(0)364 1096 y Fp(\))i Fg(i.e.)g Fp(,)g(from)f(eq.\(86\),)h(that)g Fn(u)e Fm(\025)g Fn(k)1038 1103 y Fj(1)1058 1096 y Fn(e)1081 1077 y Fi(\000j)p Fl(\025)p Fi(j)1149 1066 y Fh(\000)p Fe(k)1189 1073 y Ff(2)1211 1096 y Fp(.)23 b(With)17 b(this)g(condition) g(it)g(is)g(easy)-13 1174 y(to)e(c)o(hec)o(k,)f(from)g(eq.\(99\),)h (that)g(also)h(the)f(other)h(running)f(couplings)h(sta)o(y)f(b)q (ounded,)h(remem)n(b)q(ering)-13 1252 y(that)g Fn(g)115 1259 y Fj(3)p Fl(;)p Fj(0)163 1252 y Fn(;)8 b(g)208 1259 y Fj(5)p Fl(;)p Fj(0)255 1252 y Fn(;)g(g)300 1259 y Fj(6)p Fl(;)p Fj(0)348 1252 y Fn(;)g(\034)391 1259 y Fj(0)410 1252 y Fn(;)g(\022)455 1259 y Fj(0)489 1252 y Fp(=)13 b Fn(O)q Fp(\()p Fn(\025u)p Fp(\).)10 1330 y(In)21 b(conclusion)g(if)g (the)g(fermions)f(are)h(spinning)g(the)g(second)g(statemen)o(t)f(of)h (the)g(main)f(theorem)-13 1409 y(should)f(hold)h(in)f(the)f(case)i(of)f (repulsiv)o(e)f(in)o(teraction)g(or)h(if)g Fm(j)p Fn(u)p Fm(j)f(\025)h Fn(k)1289 1416 y Fj(1)1309 1409 y Fn(e)1332 1390 y Fi(\000j)p Fl(\025)p Fi(j)1400 1379 y Fh(\000)p Fe(k)1440 1386 y Ff(2)1461 1409 y Fp(.)30 b(In)19 b(order)g(to)g(pro)o (v)o(e)-13 1487 y(this)g(one)h(has)g(to)f(rep)q(eat)h(the)f(analysis)g (of)h(the)f(preceding)g(sections)g(and)h(to)f(sho)o(w)h(that)g(the)f (\015o)o(w)-13 1565 y(generated)f(b)o(y)f(the)h(complete)d(b)q(eta)k (function)f(qualitativ)o(ely)d(coincides)i(with)h(the)f(\015o)o(w)i (generated)-13 1643 y(b)o(y)c(the)g(b)q(eta)h(function)f(truncated)g (to)h(the)f(second)g(order)h(eq.\(99\),)f(exploiting)f(some)g (cancellations)-13 1722 y(whic)o(h)19 b(should)i(o)q(ccur)g(in)f(the)g (Beta)f(function.)33 b(This)21 b(seems)e(not)h(to)q(o)i(di\016cult)d (and)h(w)o(e)g(hop)q(e)h(to)-13 1800 y(discuss)16 b(it)g(in)g(an)h (other)f(pap)q(er.)10 1878 y(In)e(the)f(spinning)g(case)h(with)f (attractiv)o(e)g(in)o(teraction)f Fn(\025)j(<)e Fp(0)h(and)g(no)g(p)q (erio)q(dic)f(p)q(oten)o(tial)h(there)f(are)-13 1956 y(no)i(rigorous)g(result)f(ab)q(out)i(the)e(Sc)o(h)o(winger)f(function) h(b)q(eha)o(viour,)h(see)f(\(B.M.\);)e(on)j(the)f(other)h(hand)-13 2035 y(in)20 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(B.C.S.)f(sup)q(erconductivit)o(y)h(in)g(solid)h(state)g(mo)q(dels)e (or)j(to)f(the)f(mass)g(generation)-13 2504 y(in)h(the)h(relativistic)d (standard)k(mo)q(del,)d(but)i(as)h(far)f(as)g(w)o(e)f(kno)o(wn)h(no)g (rigorous)h(results)f(are)f(kno)o(wn)-13 2582 y(\(although)h(new)f (insigth)g(w)o(as)h(giv)o(en)f(in)f(\(F.T.M.R.\)\).)907 2728 y(34)p eop %%Page: 35 35 35 34 bop -13 -5 a Fo(3)110 b(App)r(endix)27 b(1)-13 123 y Fp(This)16 b(App)q(endix)f(is)g(dev)o(oted)g(to)h(the)g(\014rst)g (statemen)o(t)e(of)i(the)f(theorem.)20 b(The)c Fg(Euclide)n(an)i (\014elds)f Fp(are)-13 201 y(here)e(de\014ned)h(as:)629 279 y Fn( )663 259 y Fl(")661 292 y(x;\033)727 279 y Fp(=)779 221 y Fk(Z)829 279 y Fn(dk)r(\036)p Fp(\()926 266 y Fn(~)929 279 y(k)r(;)7 b(~)-23 b(x)p Fp(\))p Fn(e)1048 259 y Fl(i"k)1094 264 y Ff(0)1111 259 y Fl(x)1131 264 y Ff(0)1151 279 y Fn( )1185 259 y Fl(")1183 292 y(k)q(;\033)-13 395 y Fp(with)17 b Fn(e)122 377 y Fl(ik)152 382 y Ff(0)169 377 y Fl(\014)207 395 y Fp(=)e Fm(\000)p 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752 y Fp(1)p 883 774 49 2 v 883 820 a Fm(N)945 727 y Fk(Z)995 786 y Fn(P)7 b Fp(\()p Fn(d )r Fp(\))p Fn(e)1153 766 y Fi(\000)1185 757 y Fj(\026)1180 766 y Fl(V)g Fj(\()p Fl( )q Fj(+)p Fl(')p Fj(\))1312 786 y Fn(:)439 b Fp(\(100\))-13 919 y(where)12 b Fm(N)21 b Fp(=)237 884 y Fk(R)273 919 y Fn(P)7 b Fp(\()p Fn(d )r Fp(\))p Fn(e)431 901 y Fi(\000)463 893 y Fj(\026)458 901 y Fl(V)h Fj(\()p Fl( )q Fj(\))553 919 y Fp(is)k(a)g(normalization)f (constan)o(t)i(so)g(that)f Fn(V)1317 926 y Fl(ef)t(f)1377 919 y Fp(\(0\))i(=)g(0)f(and)1640 907 y(\026)1633 919 y Fn(V)e Fp(\()p Fn( )r Fp(\))i(=)h Fn(\025V)-13 998 y Fp(where:)256 1076 y Fn(V)25 b Fp(=)361 1034 y Fk(X)360 1127 y Fl(\033)o(;\033)410 1117 y Fh(0)431 1017 y Fk(Z)454 1112 y Fj(\003)p Fi(\002)p Fj(\003)540 1076 y Fn(dx)593 1083 y Fj(1)613 1076 y Fn(dx)666 1083 y Fj(2)686 1076 y Fn(v)r Fp(\()o Fn(~)-23 b(x)759 1083 y Fj(1)789 1076 y Fm(\000)10 b Fn(~)-23 b(x)867 1083 y Fj(2)886 1076 y Fp(\))p Fn(\016)r Fp(\()p Fn(x)976 1083 y Fj(0)p Fl(;)p Fj(1)1033 1076 y Fm(\000)11 b Fn(x)1111 1083 y Fj(0)p Fl(;)p Fj(2)1158 1076 y Fp(\))p Fn( )1211 1055 y Fj(+)1209 1088 y Fl(x)1229 1093 y Ff(1)1246 1088 y Fl(;\033)1279 1076 y Fn( )1313 1055 y Fj(+)1311 1089 y Fl(x)1331 1094 y Ff(2)1348 1089 y Fl(;\033)1379 1079 y Fh(0)1392 1076 y Fn( )1426 1055 y Fi(\000)1424 1089 y Fl(x)1444 1094 y Ff(2)1461 1089 y Fl(;\033)1492 1079 y Fh(0)1506 1076 y Fn( )1540 1055 y Fi(\000)1538 1088 y Fl(x)1558 1093 y Ff(1)1575 1088 y Fl(;\033)1765 1076 y Fp(\(101\))-13 1206 y(with)16 b(\003)d(=)h(\()p Fm(\000)p Fn(\014)s(=)p Fp(2)p Fn(;)8 b(\014)s(=)p Fp(2\))i Fm(\002)h Fp(\()p Fm(\000)p Fn(N)q(=)p Fp(2)p Fn(;)d(N)q(=)p Fp(2\).)23 b(As)15 b Fn(u)h Fp(is)g(not)h(small)d(the)i(p)q(erio)q(dic)g(p)q(oten) o(tial)f(has)i(to)f(b)q(e)-13 1284 y(considered)g(as)g(a)h(part)g(of)f (the)g(free)g(hamiltonian.)10 1362 y(The)h(t)o(w)o(o)f(p)q(oin)o(ts)g (Sc)o(h)o(winger)g(function)g(is)g(giv)o(en)f(in)h(term)f(of)h(the)g (e\013ectiv)o(e)e(p)q(oten)o(tial)i(b)o(y:)419 1519 y Fn(S)s Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)g Fn(g)r Fp(\()p Fn(x;)8 b(y)r Fp(\))j(+)829 1460 y Fk(Z)879 1519 y Fn(dz)r(dz)979 1498 y Fi(0)991 1519 y Fn(g)r Fp(\()p Fn(x;)d(z)r Fp(\))p Fn(V)1157 1526 y Fj(2)1176 1519 y Fp(\()p Fn(z)r(;)g(z)1267 1498 y Fi(0)1279 1519 y Fp(\))p Fn(g)r Fp(\()p Fn(z)1367 1498 y Fi(0)1378 1519 y Fn(;)g(y)r Fp(\))320 b(\(102\))-13 1635 y(with)17 b Fn(V)127 1642 y Fj(2)147 1635 y Fp(\()p Fn(z)r(;)8 b(z)238 1617 y Fi(0)249 1635 y Fp(\))16 b(=)352 1611 y Fl(@)373 1600 y Ff(2)389 1611 y Fl(V)410 1616 y Ff(2)427 1611 y Fj(\()p Fl(\036)p Fj(\))p 343 1623 142 2 v 343 1656 a Fl(@)r(')387 1641 y Ff(+)387 1661 y Fe(z)412 1656 y Fl(@)r(')456 1641 y Fh(\000)456 1672 y Fe(z)472 1665 y Fh(0)489 1635 y Fp(.)25 b(It)17 b(is)g(con)o(v)o (enien)o(t)f(to)h(rewrite)g(the)g(grassmanian)h(in)o(tegral)e(in)i (eq.\(100\))-13 1713 y(as:)530 1841 y Fn(e)553 1820 y Fi(\000)p Fl(V)601 1826 y Fe(ef)s(f)653 1820 y Fj(\()p Fl(')p Fj(\))719 1841 y Fp(=)788 1807 y(1)p 776 1829 49 2 v 776 1875 a Fm(N)837 1782 y Fk(Z)887 1841 y Fn(P)7 b Fp(\()p Fn(d )1001 1848 y Fl(i:r)o(:)1050 1841 y Fp(\))p Fn(e)1092 1820 y Fi(\000)p Fl(V)1147 1809 y Ff(0)1165 1820 y Fj(\()p Fl( )1202 1825 y Fe(i:r)o(:)1247 1820 y Fj(+)p Fl(')p Fj(\))1765 1841 y Fp(\(103\))530 1950 y Fn(e)553 1930 y Fi(\000)p Fl(V)608 1918 y Ff(0)625 1930 y Fj(\()p Fl( )662 1935 y Fe(i:r)o(:)708 1930 y Fj(+)p Fl(')p Fj(\))788 1950 y Fp(=)840 1892 y Fk(Z)890 1950 y Fn(P)g Fp(\()p Fn(d )1004 1957 y Fl(u:v)q(:)1064 1950 y Fp(\))p Fn(e)1106 1930 y Fi(\000)1138 1921 y Fj(\026)1133 1930 y Fl(V)h Fj(\()p Fl( )1199 1935 y Fe(i:r)o(:)1244 1930 y Fj(+)p Fl( )1294 1934 y Fe(u:v)q(:)1350 1930 y Fj(+)p Fl(')p Fj(\))1765 1950 y Fp(\(104\))-13 2078 y(where)21 b Fn( )167 2060 y Fl(u:v)q(:)227 2078 y Fp(,)i Fn( )298 2060 y Fl(i:r)o(:)347 2078 y Fp(,)g Fn(')f Fp(are)g(an)o(ticomm)n (uting)d(grassmanian)k(\014elds,)g Fn(P)7 b Fp(\()p Fn(d )1414 2060 y Fl(u:v)q(:)1474 2078 y Fp(\),)23 b Fn(P)7 b Fp(\()p Fn(d )1646 2060 y Fl(i:r)o(:)1695 2078 y Fp(\))22 b(denote)-13 2157 y(resp)q(ectiv)o(ely)11 b(the)j(grassmanian)h(in)o(tegrations)e (with)h(v)m(anishing)h(cross)f(propagator)i(and)f(with)f(prop-)-13 2235 y(agators)k Fn(g)181 2242 y Fl(u:v)q(:)242 2235 y Fn(andg)345 2242 y Fl(i:r)o(:)410 2235 y Fp(giv)o(en)d(b)o(y:)234 2375 y Fn(g)257 2382 y Fl(i:r)o(:)306 2375 y Fp(\()p Fn(x;)8 b(y)r Fp(\))41 b(=)540 2317 y Fk(Z)590 2375 y Fn(dk)666 2342 y(e)689 2323 y Fi(\000)p Fl(ik)746 2328 y Ff(0)764 2323 y Fj(\()p Fl(x)798 2328 y Ff(0)815 2323 y Fi(\000)p Fl(y)859 2328 y Ff(0)877 2323 y Fj(\))p 647 2364 265 2 v 647 2417 a Fm(\000)p Fn(ik)728 2424 y Fj(0)759 2417 y Fp(+)11 b Fn(E)s Fp(\()863 2404 y Fn(~)866 2417 y(k)r Fp(\))917 2375 y Fn(\036)p Fp(\()962 2362 y Fn(~)965 2375 y(k)r(;)d Fm(\000)o Fn(~)-23 b(x)o Fp(\))p Fn(\036)p Fp(\()1144 2362 y Fn(~)1147 2375 y(k)r(;)8 b(~)-24 b(y)q Fp(\)\(1)12 b Fm(\000)f Fn(h)p Fp(\()p Fn(k)1419 2355 y Fj(2)1417 2388 y(0)1449 2375 y Fp(+)g Fn(E)s Fp(\()1553 2362 y Fn(~)1556 2375 y(k)r Fp(\))1602 2355 y 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Fl(;\033)1114 1305 y Ff(1)1141 1288 y Fm(\001)h(\001)g(\001)h Fn( )1242 1268 y Fi(\000)1240 1300 y Fl(x)1260 1305 y Ff(2)p Fe(n)1296 1300 y Fl(;\033)1326 1305 y Ff(2)p Fe(n)1364 1288 y Fn(W)1410 1295 y Fl(n)1434 1288 y Fp(\()p Fn(x)1481 1295 y Fj(1)1500 1288 y Fn(;)f(:)g(:)g(:)g(;)g (x)1638 1295 y Fj(2)p Fl(n)1678 1288 y Fp(;)g Fn(\025)p Fp(\))-13 1431 y(where)16 b Fn(K)t Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b(=)g Fn(v)r Fp(\()p Fn(x)e Fm(\000)f Fn(y)r Fp(\))p Fn(R)p Fp(\()p Fn(y)r(;)e(y)r Fp(\),)15 b(the)h(k)o(ernels)f Fn(W)1000 1438 y Fl(n)1040 1431 y Fp(are)h(pro)q(ducts)h(of)g(suitable) f(delta)g(functions)-13 1509 y(times)f(b)q(ounded)k(functions)e (analytic)g(in)h Fn(\025)g Fp(if)f Fm(j)p Fn(\025)p Fm(j)f(\024)f Fn(")j Fp(and,)g(if)f Fn(d)p Fp(\()p Fn(x)1263 1516 y Fj(1)1291 1509 y Fn(:)8 b(:)g(:)f(;)h(x)1406 1516 y Fl(n)1430 1509 y Fp(\))17 b(is)g(the)h(length)f(of)h(the)-13 1587 y(shortest)i(tree)e(connecting)h(the)g(p)q(oin)o(ts)h(\(\\tree)f 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Fk(X)548 2106 y Fl(h)596 1996 y Fk(Z)646 2055 y Fn(d)671 2034 y Fj(2)691 2055 y Fn(k)723 2021 y(f)5 b Fp(\()p Fn(\015)799 2003 y Fi(\000)p Fj(2)p Fl(h)867 2021 y Fp(\()p Fn(k)913 2003 y Fj(2)911 2033 y(0)944 2021 y Fp(+)11 b(\()p Fn(")p Fp(\()1051 2008 y Fn(~)1054 2021 y(k)q Fp(\))g Fm(\000)g Fn(\026)p Fp(\))1208 2003 y Fj(2)1228 2021 y Fp(\)\))p Fn(\036)p Fp(\()1311 2008 y Fn(~)1314 2021 y(k)r(;)c(~)-23 b(x)p Fp(\))p Fn(\036)p Fp(\()p Fm(\000)1494 2008 y Fn(~)1497 2021 y(k)q(;)8 b(~)-24 b(y)r Fp(\))p Fn(e)1613 2003 y Fl(ik)1643 2008 y Ff(0)1660 2003 y Fj(\()p Fl(x)1694 2008 y Ff(0)1711 2003 y Fi(\000)p Fl(y)1755 2008 y Ff(0)1773 2003 y Fj(\))p 723 2043 1066 2 v 1087 2097 a Fm(\000)p Fn(ik)1168 2104 y Fj(0)1198 2097 y Fp(+)11 b Fn(")p Fp(\()1286 2084 y Fn(~)1289 2097 y(k)r Fp(\))g Fm(\000)g Fn(\026)1808 2055 y Fp(=)468 2241 y(=)566 2188 y Fj(0)545 2200 y Fk(X)520 2292 y Fl(h)p Fj(=)p Fi(\0001)638 2183 y Fk(Z)688 2241 y Fn(d)713 2221 y Fj(2)733 2241 y Fn(k)r(\036)p Fp(\()805 2228 y 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(is)h(at)g(the)g(cen)o(ter)e(of)-13 2525 y(the)i(gap)i(w)o(e)f(ha)o(v)o (e)f(that)h Fn(")p Fp(\()510 2512 y Fn(~)513 2525 y(k)r Fp(\))13 b Fm(\000)h Fn(\026)20 b Fp(is)g(nev)o(er)e(v)m(anishing)j(so) f(that)h(only)e(a)i(\014nite)e(n)o(um)o(b)q(er)f(of)i(term)-13 2603 y Fn(g)12 2585 y Fj(\()p Fl(h)p Fj(\))62 2603 y Fp(\()p Fn(k)r Fp(\))15 b(con)o(tributes)g(to)h Fn(g)477 2610 y Fl(i:r)o(:)526 2603 y Fp(\()p Fn(k)r Fp(\).)21 b(W)l(e)15 b(call)f Fn(h)825 2585 y Fi(\003)860 2603 y Fp(the)h(last)h Fn(h)f Fp(suc)o(h)h(that)f Fn(f)5 b Fp(\()p Fn(\015)1368 2585 y Fi(\000)p Fj(2)p Fl(h)1436 2603 y Fp(\()p Fn(k)1482 2585 y Fj(2)1480 2615 y(0)1511 2603 y Fp(+)k(\()p Fn(")p Fp(\()1616 2590 y Fn(~)1619 2603 y(k)r Fp(\))g Fm(\000)g Fn(\026)p Fp(\))1770 2585 y Fj(2)1790 2603 y Fp(\)\))16 b(is)907 2728 y(36)p eop %%Page: 37 37 37 36 bop -13 -5 a Fp(not)16 b(iden)o(tically)d(v)m(anishing)j(and)g (do)q(es)g(not)g(con)o(tain)g Fn(\031)r(=a)f Fp(in)g(its)g(supp)q(ort.) 22 b(W)l(e)16 b(call,)e(for)i(simplicit)n(y)l(,)-13 74 y Fn(g)12 55 y Fl(h)32 44 y Fh(\003)50 55 y Fi(\000)p Fj(1)97 74 y Fp(\()p Fn(k)r Fp(\))22 b(the)g(sum)f(of)h(the)f Fn(g)558 55 y Fl(h)581 74 y Fp(\()p Fn(k)r Fp(\))h(whose)g(supp)q(ort)h (con)o(tain)f Fn(\031)r(=a)p Fp(.)37 b(W)l(e)22 b(note)g(that)g([)p Fn(")p Fp(\(\(2)p Fn(\031)r Fp(\))1775 55 y Fj(+)1804 74 y Fp(\))15 b Fm(\000)-13 152 y Fn(\026)p Fp(])p Fn(\015)58 134 y Fl(h)78 122 y Fh(\003)96 134 y Fi(\000)p Fj(1)157 152 y Fp(=)f Fn(O)q Fp(\(\001\))i(where)g(\001)e(=)f([)p Fn(")p Fp(\()p Fn(\031)675 134 y Fj(+)704 152 y Fp(\))e(+)g Fn(")p Fp(\()p Fn(\031)855 134 y Fi(\000)884 152 y Fp(\)])p Fn(=)p Fp(2)16 b(is)g(the)g(gap)h(amplitude.)10 230 y(The)g(follo)o (wing)f(lemm)o(a)e(holds:)-13 356 y Fd(Lemm)o(a)i(3.1)24 b Fg(F)l(or)d(any)f(inte)n(ger)i Fn(N)k Fg(and)21 b(for)f(any)h Fn(u)f Fg(such)i(that)f Fp(\001)e Fm(6)p Fp(=)h(0)h Fg(ther)n(e)g (exist)h(c)n(ontinuous)-13 435 y(functions)d Fn(C)234 442 y Fl(N)267 435 y Fp(\()p Fn(u)p Fp(\))e 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b(in)f(the)g(supp)q(ort)i(of)f Fn(g)1335 1297 y Fl(h)1357 1315 y Fp(\()p Fn(k)r Fp(\),)g(are)f(b)q(ounded)i(for)f(an) o(y)-13 1393 y Fn(N)21 b Fp(and)c(suitable)f(con)o(tin)o(uous)g (functions)g Fn(C)813 1400 y Fl(N)847 1393 y Fp(\()p Fn(u)p Fp(\))g(b)o(y:)153 1550 y Fm(j)p Fn(@)196 1529 y Fl(N)191 1560 y(~)193 1568 y(k)228 1550 y Fp([)p Fn(\036)p Fp(\()287 1537 y Fn(~)290 1550 y(k)r(;)340 1536 y Fp(\026)338 1550 y Fn(~)-23 b(x)o Fp(\))p Fn(\036)p Fp(\()p Fm(\000)469 1537 y Fn(~)472 1550 y(k)r(;)521 1536 y Fp(\026)521 1550 y Fn(~)f(y)q Fp(\)])p Fm(j)14 b(\024)664 1516 y Fn(C)699 1523 y Fl(N)733 1516 y Fp(\()p Fn(u)p Fp(\))p 664 1538 135 2 v 690 1584 a Fn(\015)718 1569 y Fl(N)t(h)1040 1550 y Fm(j)p Fn(@)1083 1529 y Fl(N)1078 1560 y(~)1080 1568 y(k)1116 1550 y Fn(")p Fp(\()1155 1537 y Fn(~)1158 1550 y(k)q Fp(\))p Fm(j)g(\024)g Fn(C)1319 1557 y Fl(N)1352 1550 y Fp(\()p Fn(u)p Fp(\))p Fn(\015)1446 1529 y Fi(\000)p Fl(h)p Fj(\()p Fl(N)t Fi(\000)p Fj(1\))1765 1550 y Fp(\(110\))-13 1665 y(F)l(or)i(the)g(last)h(propagator)h Fn(g)525 1647 y Fj(\()p Fl(h)559 1636 y Fh(\003)577 1647 y Fi(\000)p Fj(1\))654 1665 y Fp(w)o(e)e(write)188 1748 y Fk(Z)238 1806 y Fn(d)263 1786 y Fj(2)283 1806 y Fn(k)315 1773 y(f)5 b Fp(\()p Fn(\015)391 1755 y Fi(\000)p Fj(2\()p Fl(h)470 1743 y Fh(\003)489 1755 y Fi(\000)p Fj(1\))550 1773 y Fp(\()p Fn(k)596 1755 y Fj(2)594 1785 y(0)626 1773 y Fp(+)11 b(\()p Fn(")p Fp(\()733 1759 y Fn(~)736 1773 y(k)r Fp(\))g Fm(\000)g Fn(\026)p Fp(\))891 1755 y Fj(2)911 1773 y Fp(\)\))p Fn(\036)p Fp(\()994 1759 y Fn(~)997 1773 y(k)r(;)c(~)-23 b(x)p Fp(\))p Fn(\036)p Fp(\()p Fm(\000)1177 1759 y Fn(~)1180 1773 y(k)q(;)8 b(~)-24 b(y)r Fp(\))p Fn(e)1296 1755 y Fl(ik)1326 1760 y Ff(0)1343 1755 y Fj(\()p Fl(x)1377 1760 y Ff(0)1394 1755 y Fi(\000)p Fl(y)1438 1760 y Ff(0)1456 1755 y Fj(\))p 315 1795 1157 2 v 724 1848 a Fm(\000)p Fn(ik)805 1855 y Fj(0)836 1848 y Fp(+)11 b Fn(")p Fp(\()924 1835 y Fn(~)927 1848 y(k)r Fp(\))g Fm(\000)f Fn(\026)1490 1806 y Fp(=)188 1901 y Fk(Z)211 1995 y Fj([)p Fi(\000)p Fl(\031)r(;\031)q Fj(])322 1960 y Fn(d)347 1939 y Fj(2)367 1960 y Fn(k)399 1926 y(f)5 b Fp(\()p Fn(\015)475 1908 y Fi(\000)p Fj(2\()p Fl(h)554 1896 y Fh(\003)572 1908 y Fi(\000)p Fj(1\))633 1926 y Fp(\()p Fn(k)679 1908 y Fj(2)677 1938 y(0)710 1926 y Fp(+)11 b(\()p Fn(")p Fp(\()817 1913 y Fn(~)820 1926 y(k)r Fp(\))g Fm(\000)f Fn(\026)p Fp(\))974 1908 y Fj(2)995 1926 y Fp(\)\))p Fn(\036)1062 1933 y Fj(0)1081 1926 y Fp(\()1097 1913 y Fn(~)1100 1926 y(k)r(;)d(~)-23 b(x)p Fp(\))p Fn(\036)1225 1933 y Fj(0)1244 1926 y Fp(\()p Fm(\000)1299 1913 y Fn(~)1302 1926 y(k)r(;)8 b(~)-24 b(y)q Fp(\))p Fn(e)1418 1908 y Fl(ik)1448 1913 y Ff(0)1466 1908 y Fj(\()p Fl(x)1500 1913 y Ff(0)1516 1908 y Fi(\000)p Fl(y)1560 1913 y Ff(0)1579 1908 y Fj(\))p 399 1948 1196 2 v 817 2002 a Fm(\000)p Fn(ik)898 2009 y Fj(0)929 2002 y Fp(+)11 b Fn(")1001 2009 y Fj(0)1021 2002 y Fp(\()1037 1989 y Fn(~)1040 2002 y(k)r Fp(\))g Fm(\000)f Fn(\026)1610 1960 y Fp(+)188 2055 y Fk(Z)211 2149 y Fj([)p Fi(\000)p Fj(2)p Fl(\031)r(;)p Fi(\000)p Fl(\031)r Fj(])p Fi([)p Fj([)p Fl(\031)q(;)p Fj(2)p Fl(\031)p Fj(])480 2113 y Fn(d)505 2093 y Fj(2)525 2113 y Fn(k)557 2079 y(f)5 b Fp(\()p Fn(\015)633 2061 y Fi(\000)p Fj(2\()p Fl(h)712 2050 y Fh(\003)731 2061 y Fi(\000)p Fj(1\))792 2079 y Fp(\()p Fn(k)838 2061 y Fj(2)836 2092 y(0)868 2079 y Fp(+)11 b(\()p Fn(")p Fp(\()975 2066 y Fn(~)978 2079 y(k)r Fp(\))g Fm(\000)g Fn(\026)p Fp(\))1133 2061 y Fj(2)1153 2079 y Fp(\)\))p Fn(\036)1220 2086 y Fj(1)1240 2079 y Fp(\()1256 2066 y Fn(~)1259 2079 y(k)r(;)c(~)-23 b(x)o Fp(\))p Fn(\036)1383 2086 y Fj(1)1403 2079 y Fp(\()p Fm(\000)1458 2066 y Fn(~)1461 2079 y(k)q(;)8 b(~)-24 b(y)r Fp(\))p Fn(e)1577 2061 y Fl(ik)1607 2066 y Ff(0)1624 2061 y Fj(\()p Fl(x)1658 2066 y Ff(0)1675 2061 y Fi(\000)p Fl(y)1719 2066 y Ff(0)1737 2061 y Fj(\))p 557 2101 V 976 2155 a Fm(\000)p Fn(ik)1057 2162 y Fj(0)1088 2155 y Fp(+)11 b Fn(")1160 2162 y Fj(1)1179 2155 y Fp(\()1195 2142 y Fn(~)1198 2155 y(k)r Fp(\))g Fm(\000)g 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2502 y Fh(\003)610 2513 y Fi(\000)p Fj(1\))671 2531 y Fp(\()p Fn(k)717 2513 y Fj(2)715 2544 y(0)748 2531 y Fp(+)11 b(\()p Fn(")p Fp(\()855 2518 y Fn(~)858 2531 y(k)q Fp(\))g Fm(\000)g Fn(\026)p Fp(\))1012 2513 y Fj(2)1032 2531 y Fp(\)\))p Fn(\036)1099 2538 y Fj(1)1119 2531 y Fp(\()1135 2518 y Fn(~)1138 2531 y(k)r(;)c(~)-23 b(x)o Fp(\))p Fn(\036)1262 2538 y Fj(1)1282 2531 y Fp(\()p Fm(\000)1337 2518 y Fn(~)1340 2531 y(k)r(;)8 b(~)-24 b(y)q Fp(\))p Fn(e)1456 2513 y Fl(ik)1486 2518 y Ff(0)1503 2513 y Fj(\()p Fl(x)1537 2518 y Ff(0)1554 2513 y Fi(\000)p Fl(y)1598 2518 y Ff(0)1616 2513 y Fj(\))p 436 2553 V 855 2607 a Fm(\000)p Fn(ik)936 2614 y Fj(0)967 2607 y Fp(+)11 b Fn(")1039 2614 y Fj(1)1058 2607 y Fp(\()1074 2594 y Fn(~)1077 2607 y(k)r Fp(\))g Fm(\000)g Fn(\026)907 2728 y Fp(37)p eop %%Page: 38 38 38 37 bop -13 -5 a Fp(so)13 b(that)g(w)o(e)g(ha)o(v)o(e)e(written)i (the)f(propagator)j(as)e(the)g(sum)e(of)i(the)g(in)o(tegrals)f(of)h(t)o (w)o(o)f(p)q(erio)q(dic)h(function)-13 74 y(o)o(v)o(er)j(a)h(p)q(erio)q (d)h(and)f(w)o(e)g(can)g(in)o(tegrate)f(b)o(y)g(parts)i(without)f(b)q (oundary)h(terms.)k(Hence)16 b(pro)q(ceeding)-13 152 y(as)h(for)f Fn(h)e(>)g(h)243 134 y Fi(\003)274 152 y Fm(\000)d Fp(1)16 b(using)h(eq.\(110\))f(w)o(e)g(ha)o(v)o(e)562 290 y Fn(g)587 270 y Fj(\()p Fl(h)621 258 y Fh(\003)639 270 y Fi(\000)p Fj(1\))700 290 y Fp(\()p Fn(x;)8 b(y)r Fp(\))13 b Fm(\024)967 257 y Fn(\015)995 238 y Fl(h)1015 227 y Fh(\003)1033 238 y Fi(\000)p Fj(1)1080 257 y Fn(C)1115 264 y Fl(N)1149 257 y Fp(\()p Fn(u)p Fp(\))p 885 279 412 2 v 885 324 a(1)e(+)g(\()p Fn(\015)1016 310 y Fl(h)1036 301 y Fh(\003)1055 310 y Fi(\000)p Fj(1)1102 324 y Fm(j)p Fn(x)f Fm(\000)h Fn(y)r Fm(j)p Fp(\))1263 310 y Fl(N)10 422 y Fp(This)17 b(ends)f(the)g(pro)q(of.)10 619 y(The)h(e\013ectiv)o (e)d(p)q(oten)o(tial)i(is)g(giv)o(en)f(b)o(y)292 752 y Fm(\000)p Fn(V)359 759 y Fl(ef)t(f)419 752 y Fp(\()p Fn(')p Fp(\))f(=)f(log)626 693 y Fk(Z)675 752 y Fn(P)7 b Fp(\()p Fn(d )789 759 y Fl(i:r)o(:)838 752 y Fp(\))p Fn(e)880 731 y Fi(\000)p Fl(V)928 736 y Ff(0)945 731 y Fj(\()p Fl( )982 736 y Fe(i:r)o(:)1028 731 y Fj(+)p Fl(')p Fj(\))1108 752 y Fp(=)1175 698 y Fi(1)1163 710 y Fk(X)1160 801 y Fl(n)p Fj(=1)1248 718 y Fp(1)p 1239 740 43 2 v 1239 786 a Fn(n)p Fp(!)1287 752 y Fm(E)1317 731 y Fl(T)1345 752 y Fp(\()p Fn(V)1392 759 y Fj(0)1412 752 y Fn(;)h(:)g(:)g(:)g(;)g(V)1550 759 y Fj(0)1364 779 y Fk(|)p 1383 779 66 5 v 66 w({z)p 1487 779 V 66 w(})1456 816 y Fl(n)1570 752 y Fp(\))14 b(=)308 855 y Fi(1)295 868 y Fk(X)292 958 y Fl(n)p Fj(=1)381 876 y Fp(1)p 372 898 43 2 v 372 944 a Fn(n)p Fp(!)459 868 y Fk(X)428 955 y Fl(\013)451 960 y Ff(1)468 955 y Fl(;:::)o(\013)530 959 y Fe(n)669 868 y Fk(X)569 959 y Fj(~)559 968 y Fl(X)588 973 y Ff(1)605 968 y Fi(2)p Fl(X)658 973 y Ff(1)675 968 y Fl(;:::)724 959 y Fj(~)714 968 y Fl(X)743 972 y Fe(n)765 968 y Fi(2)p Fl(X)818 972 y Fe(n)847 851 y Fk(Z)897 909 y Fn(dX)962 916 y Fj(1)991 909 y Fm(\001)8 b(\001)g(\001)g Fn(dX)1122 916 y Fl(n)1146 909 y 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Fn(x;)8 b(y)r Fp(\))13 b Fm(\024)h Fn(\015)1827 2583 y Fi(\000)p Fl(h)907 2728 y Fp(38)p eop %%Page: 39 39 39 38 bop -13 -5 a Fp(w)o(e)16 b(ha)o(v)o(e)f(that)i(eq.\(112\))f(is)g (b)q(ounded)h(b)o(y:)359 89 y Fi(1)347 102 y Fk(X)344 192 y Fl(n)p Fj(=1)423 109 y Fp(\()p Fn(C)t(\025)p Fp(\))528 91 y Fl(n)p 423 132 129 2 v 466 177 a Fn(n)p Fp(!)565 102 y Fk(X)574 194 y Fl(T)595 198 y Fe(n)645 102 y Fk(Y)633 194 y Fl(l)p Fi(2)p Fl(T)689 198 y Fe(n)718 89 y Fl(h)738 77 y Fh(\003)756 89 y Fi(\000)p Fj(1)730 102 y Fk(X)727 194 y Fl(h)p Fj(=1)896 109 y Fn(\015)924 91 y Fl(h)947 109 y Fn(C)982 116 y Fl(N)1015 109 y Fp(\()p Fn(u)p Fp(\))p 815 132 349 2 v 815 177 a(1)11 b(+)g(\()p Fn(\015)946 163 y Fl(h)968 177 y Fm(j)p Fn(x)g Fm(\000)g Fn(y)r Fm(j)p Fp(\))1130 163 y Fl(N)1211 89 y(h)1231 77 y Fh(\003)1250 89 y Fi(\000)p Fj(1)1223 102 y Fk(X)1176 194 y Fl(h)1196 199 y Ff(2)1214 194 y Fl(:::)o(h)1263 198 y Fe(n)1285 194 y Fj(=1)1338 143 y Fn(\015)1366 123 y Fi(\000)1399 94 y Fk(P)1443 104 y 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Fp(\))1472 1723 y Fe(n)p Fh(\000)p Ff(1)p 1472 1730 59 2 v 1493 1750 a(2)-13 1841 y Fp(for)16 b(suitable)g(com)o(binatorial)f(co)q (e\016cien)o(t)f Fn(K)819 1848 y Fl(n)843 1841 y Fp(.)22 b(F)l(rom)15 b(lemm)o(a)f(2)i(it)g(follo)o(ws)g(that:)594 1903 y Fk(\014)594 1928 y(\014)594 1953 y(\014)594 1978 y(\014)594 2002 y(\014)613 1944 y Fn(@)642 1926 y Fl(N)675 1944 y Fn(")p Fp(\()p Fn(k)r Fp(\))p 613 1966 150 2 v 624 2012 a(\()p Fn(@)s(k)r Fp(\))718 1997 y Fl(N)768 1903 y Fk(\014)768 1928 y(\014)768 1953 y(\014)768 1978 y(\014)768 2002 y(\014)795 1978 y Fm(\024)992 1944 y Fn(C)1027 1951 y Fl(N)1060 1944 y Fp(\()p Fn(u)p Fp(\))p 853 1966 412 2 v 853 2021 a(\(\()p Fn(k)d Fm(\000)e Fn(\031)r Fp(\))1028 2006 y Fj(2)1058 2021 y Fp(+)g(\001)1148 2006 y Fj(2)1167 2021 y Fp(\))1191 1988 y Fe(N)s Fh(\000)p Ff(1)p 1191 1995 66 2 v 1217 2015 a(2)-13 2124 y Fp(where)16 b Fn(C)163 2131 y Fl(N)196 2124 y Fp(\()p Fn(u)p Fp(\))g(is)g(a)h(con)o (tin)o(uous)f(function)g(of)h Fn(u)f 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Fj(1)1507 901 y Fp(\()p Fn(x;)1574 888 y(~)1576 901 y(k)q Fp(\))g(is)g(b)q(ounded)-13 979 y(b)o(y)129 959 y Fj(1)p 64 967 146 2 v 64 996 a Fl(h)84 1001 y Ff(0)102 996 y Fj(\()p Fl(u)p Fj(\))150 987 y Fe(n)p Fh(\000)p Ff(1)236 979 y Fp(b)q(ecause)i(they)e(are)i (analytic)e(in)h Fn(q)r Fp(.)35 b(The)21 b Fn(n)p Fp(-th)h(deriv)m (ativ)o(e)d(of)j Fn(f)1484 961 y Fi(\006)1535 979 y Fp(are)f(b)q (ounded)h(b)o(y)-13 1062 y Fn(C)26 1044 y Fl(n)49 1062 y Fn(h)77 1069 y Fj(0)97 1062 y Fp(\()p Fn(u)p Fp(\))163 1044 y Fi(\000)p Fl(n)213 1062 y Fp(.)28 b(The)18 b(only)g(term)f(to)h (con)o(trol)g(is)863 1012 y Fk(q)p 905 1012 162 2 v 50 x Fn(h)933 1069 y Fj(0)953 1062 y Fp(\()p Fn(u)p Fp(\))p Fn(=q)q Fp(.)28 b(W)l(e)18 b(ha)o(v)o(e)f(that)i(the)f Fn(n)p Fp(-th)h(deriv)m(ativ)o(e)d(of)-13 1140 y(this)g(function)g(is)g (giv)o(en)f(b)o(y:)443 1225 y Fk(X)458 1312 y Fl(m)626 1225 y Fk(X)511 1314 y Fl(n)532 1319 y Ff(1)550 1314 y Fj(+)p Fl(n)598 1319 y Ff(2)616 1314 y Fj(+)p Fi(\001\001\001)p Fj(+)p 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y Fi(\000)p Fj(\()p Fl(n)p Fi(\000)p Fj(1\))-13 1478 y Fp(w)o(e)16 b(get)g(that)h Fn(d)271 1460 y Fl(n)303 1430 y Fk(\020)328 1428 y(q)p 370 1428 162 2 v 50 x Fn(h)398 1485 y Fj(0)418 1478 y Fp(\()p Fn(u)p Fp(\))o Fn(=q)531 1430 y Fk(\021)564 1478 y Fn(=d)610 1465 y(~)613 1478 y(k)640 1460 y Fl(n)679 1478 y Fm(\024)d Fn(K)773 1485 y Fl(n)797 1478 y Fn(h)825 1485 y Fj(0)844 1478 y Fp(\()p Fn(u)p Fp(\))910 1460 y Fi(\000)p Fl(n)961 1478 y Fp(.)22 b(All)15 b(together)i(this)f (implies)e Fn(d)1556 1460 y Fl(n)1580 1478 y Fn(\036)p Fp(\()1625 1465 y Fn(~)1628 1478 y(k)r(;)7 b(~)-23 b(x)o Fp(\))p Fn(=d)1769 1465 y(~)1772 1478 y(k)1799 1460 y Fl(n)1838 1478 y Fm(\024)-13 1557 y Fn(K)32 1538 y Fi(0)28 1569 y Fl(n)52 1557 y Fn(h)80 1564 y Fj(0)100 1557 y Fp(\()p Fn(u)p Fp(\))166 1538 y Fi(\000)p Fl(n)216 1557 y Fp(.)10 1635 y(If)55 1622 y Fn(~)58 1635 y(k)10 b Fm(\000)e Fn(\031)16 b(>)d(O)q Fp(\(\001\))j(it)e(is)h(enough)g(to)h(observ)o(e)e (that)h(the)g(deriv)m(ativ)o(es)f(of)1343 1585 y Fk(q)p 1384 1585 197 2 v 1384 1635 a Fn(q)f Fp(+)e Fn(h)1496 1642 y Fj(0)1516 1635 y Fp(\()p Fn(u)p Fp(\))o Fn(=)1605 1604 y Fm(p)p 1647 1604 24 2 v 31 x Fn(q)17 b Fp(are)d(giv)o(en)-13 1713 y(b)o(y:)283 1758 y Fk(X)201 1847 y Fl(m;m)273 1852 y Ff(1)290 1847 y Fj(+)p Fl(m)348 1852 y Ff(2)366 1847 y Fj(=)p Fl(m)515 1758 y Fk(X)433 1847 y Fl(n)454 1852 y Ff(1)472 1847 y Fj(+)p Fi(\001\001\001)p Fj(+)p Fl(n)577 1851 y Fe(m)607 1847 y Fj(=)p Fl(n)664 1799 y Fn(K)709 1779 y Fl(m;m)781 1784 y Ff(1)799 1779 y Fl(;m)840 1784 y Ff(2)705 1812 y Fl(n)726 1817 y Ff(1)744 1812 y Fl(;:::)o(;n)814 1816 y Fe(m)880 1745 y Fl(m)869 1758 y Fk(Y)867 1849 y Fl(i)p Fj(=1)940 1766 y Fn(d)965 1748 y Fl(n)986 1753 y Fe(i)1002 1766 y Fn(q)p 938 1788 90 2 v 938 1841 a(d)960 1828 y(~)963 1841 y(k)990 1827 y Fl(n)1011 1832 y Fe(i)1032 1799 y Fn(q)1056 1779 y Fi(\000)p Fj(\()p Fl(m)1128 1784 y Ff(1)1145 1779 y Fj(+1)p Fl(=)p Fj(2\))1241 1799 y Fp(\()p Fn(q)e Fm(\000)f Fn(h)1372 1806 y Fj(0)1392 1799 y Fp(\()p Fn(u)p Fp(\)\))1477 1779 y Fi(\000)p Fj(\()p Fl(m)1549 1784 y Ff(2)1566 1779 y Fi(\000)p Fj(1)p Fl(=)p Fj(2\))-13 1915 y Fp(where)20 b Fn(K)177 1897 y Fl(m;m)249 1902 y Ff(1)266 1897 y Fl(;m)307 1902 y Ff(2)173 1927 y Fl(n)194 1932 y Ff(1)212 1927 y Fl(;:::)o(;n)282 1931 y Fe(m)347 1915 y Fp(is)g(a)h(suitable)f(com)o(binatorial)e(factor.)34 b(The)20 b(estimates)f(follo)o(ws)h(remem)n(b)q(ering)-13 1993 y(that)c Fn(d)117 1975 y Fl(n)141 1993 y Fn(q)r(=d)211 1980 y(~)214 1993 y(k)241 1975 y Fl(n)279 1993 y Fm(\024)d Fn(C)t Fp(\()p Fm(j)p Fn(k)r Fm(j)e(\000)f Fn(\031)r Fp(\))553 1975 y Fi(\000)p Fj(\()p Fl(n)p Fi(\000)p Fj(1\))676 1993 y Fp(.)10 2071 y(If)j Fn(u)g Fp(is)g(not)g(near)g(to)h(one)f(of)g (the)g Fn(u)635 2053 y Fi(\003)667 2071 y Fp(it)g(is)g(enough)g(to)h (observ)o(e)e(that)i Fn(\036)p Fp(\()1298 2058 y Fn(~)1301 2071 y(k)r(;)7 b(~)-23 b(x)o Fp(\))13 b(is)g(an)g(analytic)g(function) -13 2150 y(of)40 2137 y Fn(~)42 2150 y(k)19 b Fp(in)d(a)g(domain)g(of)g (radius)h Fn(\021)583 2157 y Fj(0)603 2150 y Fp(\()p Fn(u)p Fp(\))f(around)h Fn(k)f Fp(=)d Fn(\031)r Fp(.)21 b(By)16 b(a)g(dimensional)f(estimate)f(w)o(e)i(get:)597 2323 y Fm(j)616 2289 y Fn(@)645 2271 y Fl(n)668 2289 y Fn(\036)p Fp(\()713 2276 y Fn(~)716 2289 y(k)r(;)7 b(~)-23 b(x)p Fp(\))p 616 2311 196 2 v 674 2365 a Fn(@)700 2352 y(~)703 2365 y(k)730 2350 y Fl(n)816 2323 y Fm(j)14 b(\024)902 2281 y Fp(sup)975 2296 y Fi(j)983 2287 y Fl(~)985 2296 y(k)q Fi(\000)p Fl(\031)q Fi(j\024)p Fl(r)1118 2281 y Fn(\036)p Fp(\()1163 2268 y Fn(~)1166 2281 y(k)r(;)7 b(~)-23 b(x)o Fp(\))p 902 2311 360 2 v 1058 2357 a Fn(r)1081 2343 y Fl(n)10 2446 y Fp(Chosing)20 b Fn(r)f Fp(=)297 2391 y Fk(q)p 338 2391 369 2 v 338 2446 a Fp(\()354 2433 y Fn(~)357 2446 y(k)13 b Fm(\000)e Fn(\031)r Fp(\))494 2432 y Fj(2)524 2446 y Fp(+)g Fn(h)601 2453 y Fj(0)621 2446 y Fp(\()p Fn(u)p Fp(\))687 2432 y Fj(2)707 2446 y Fn(=)p Fp(2)19 b(and)g(noting)g(that)g(sup)1207 2462 y Fi(j)1215 2453 y Fl(~)1217 2462 y(k)q Fi(\000)p Fl(\031)q Fi(j\024)p Fl(r)1349 2446 y Fn(\036)p Fp(\()1394 2433 y Fn(~)1397 2446 y(k)r(;)7 b(~)-23 b(x)o Fp(\))19 b(is)f(con)o(tin)o (uous)h(the)-13 2525 y(second)d(of)h(the)f(b)q(ounds)h(eq.\(110\))g (follo)o(ws.)10 2603 y(Clearly)f(ab)q(o)o(v)o(e)g(analysis)g(can)h(b)q (e)f(rep)q(eated)g(in)g(corresp)q(ondence)h(of)f(eac)o(h)g(of)g(the)g (p)q(oin)o(t)h Fn(")1709 2610 y Fl(n)1732 2603 y Fp(.)907 2728 y(44)p eop %%Page: 45 45 45 44 bop -13 -5 a Fo(5)83 b(App)r(endix)26 b(3)-13 123 y Fp(In)16 b(this)g(app)q(endix)g(w)o(e)g(pro)q(of)h(lemmas)d(2.1)i (and)h(2.2.)10 201 y Fg(Pr)n(o)n(of)c(of)i(lemma)g(2.1)p Fp(:)k(The)14 b(b)q(ound)g(on)g Fn(g)780 183 y Fl(h)778 214 y(~)-18 b(!)q(;)p Fi(\000)p Fl(~)g(!)864 201 y Fp(\()p Fn(x)5 b Fm(\000)g Fn(y)r Fp(\))11 b(follo)o(ws)i(infact)g(from)f (eq.\(67\))h(b)o(y)g(in)o(tegrating)-13 279 y(b)o(y)j(parts.)10 358 y(W)l(e)g(can)h(write)e Fn(g)333 340 y Fl(h)331 370 y(~)-18 b(!)r(;~)g(!)390 358 y Fp(\()p Fn(x)11 b Fm(\000)g Fn(y)r Fp(\))k(as)i(a)g(term)d(indep)q(enden)o(t)i(on)h Fn(\033)1148 365 y Fl(h)1186 358 y Fp(plus)f(a)h(remainder:)102 497 y Fn(g)127 477 y Fl(h)125 510 y(~)-18 b(!)r(;~)g(!)184 497 y Fp(\()p Fn(x)11 b Fm(\000)g Fn(y)r Fp(\))i(=)h Fn(g)427 477 y Fl(h)425 510 y(~)-18 b(!)q Fj(;)p Fl(\033)478 516 y Fe(h)498 510 y Fj(=0)545 497 y Fp(\()p Fn(k)r Fp(\))11 b(+)670 439 y Fk(Z)712 452 y Fj(1)694 533 y(0)740 497 y Fn(dt)797 464 y(@)p 788 486 47 2 v 788 532 a(@)s(t)839 497 y(g)864 477 y Fl(h)862 510 y(~)-18 b(!)888 497 y Fp(\()p Fn(x)10 b Fm(\000)h Fn(y)r Fp(\))p Fm(j)1054 504 y Fl(t\033)1087 510 y Fe(h)1122 497 y Fp(=)j Fn(g)1199 477 y Fl(h)1197 510 y(~)-18 b(!)r Fj(;)p Fl(\033)1251 516 y Fe(h)1271 510 y Fj(=0)1318 497 y Fp(\()p Fn(k)r Fp(\))11 b(+)g Fn(C)1478 504 y Fj(2)1498 497 y Fp(\()p Fn(x)f Fm(\000)h Fn(y)r Fp(\))115 b(\(128\))-13 624 y(where)59 837 y Fn(C)94 844 y Fj(2)114 837 y Fp(\()p Fn(x)10 b Fm(\000)h Fn(y)r Fp(\))j(=)331 779 y Fk(Z)381 837 y Fn(dtdk)r(Z)509 844 y Fl(h)532 837 y Fp(\()p Fn(k)r Fp(\))597 817 y Fi(\000)p Fj(1)644 837 y Fn(\033)672 844 y Fl(h)695 837 y Fp(\()p Fn(k)r Fp(\))760 817 y Fj(2)779 837 y Fn(e)802 817 y Fl(ik)q Fj(\()p Fl(x)p Fi(\000)p Fl(y)q Fj(\))1114 804 y Fm(\000)p Fn(ik)1195 811 y Fj(0)1225 804 y Fp(+)1272 791 y Fn(~)1274 804 y(k)1301 786 y Fj(2)1332 804 y Fp(+)d(2)p Fn(!)r(\031)1464 791 y(~)1467 804 y(k)p 934 826 742 2 v 934 841 a Fk(\022)964 853 y(\020)989 901 y Fm(\000)p Fn(ik)1070 908 y Fj(0)1100 901 y Fp(+)1147 888 y Fn(~)1149 901 y(k)1176 887 y Fj(2)1196 853 y Fk(\021)1221 865 y Fj(2)1252 901 y Fm(\000)1302 853 y Fk(\020)1336 882 y Fl(\031)1355 873 y(~)1357 882 y(k)p 1331 890 50 2 v 1331 918 a(ma)1386 853 y Fk(\021)1411 862 y Fj(2)1442 901 y Fm(\000)g Fn(t\033)1538 908 y Fl(h)1559 901 y Fp(\()p Fn(k)r Fp(\))1624 841 y Fk(\023)1655 852 y Fj(2)1680 837 y Fn(:)71 b Fp(\(129\))-13 1026 y(The)16 b(b)q(ound)h(on)g Fn(C)340 1033 y Fj(2)376 1026 y Fp(follo)o(ws)f(b)o(y)g(a)h(standard)g (in)o(tegration)f(b)o(y)g(parts.)10 1222 y Fg(Pr)n(o)n(of)d(of)h(lemma) h(2.2)p Fp(:)k(W)l(e)13 b(pro)q(ceed)f(inductiv)o(ely)e(b)o(y)j (assuming)f(that)h(eq.\(79\),\(80\),\(83\),\(82\))g(hold)-13 1300 y(for)j(0)e Fm(\025)g Fn(k)i Fm(\025)d Fn(h)k Fp(and)f(pro)o(ving) h(that)f(their)g(v)m(alidit)o(y)e(for)j Fn(h)11 b Fm(\000)f Fp(1.)22 b(F)l(rom)15 b(eq.\(75\))h(and)g(the)g(analyticit)o(y)-13 1379 y(of)g(the)g(b)q(eta)h(function)f(w)o(e)g(ha)o(v)o(e)f(that:)491 1505 y Fn(Z)524 1512 y Fl(k)545 1505 y Fp(\(1)d(+)f(2)p Fn(c)694 1512 y Fj(3)714 1505 y Fn(\014)742 1512 y Fj(1)761 1505 y Fn(\025)789 1485 y Fj(2)809 1505 y Fp(\))j Fn(<)g(Z)927 1512 y Fl(k)q Fi(\000)p Fj(1)1008 1505 y Fn(<)f Fp(\(1)f(+)f Fn(c)1184 1512 y Fj(4)1204 1505 y Fn(\014)1232 1512 y Fj(1)1251 1505 y Fn(\025)1279 1485 y Fj(2)1299 1505 y Fp(\))p Fn(Z)1351 1512 y Fl(k)-13 1632 y Fp(where)16 b Fn(c)149 1639 y Fj(1)168 1632 y Fn(;)8 b(c)211 1639 y Fj(2)247 1632 y Fp(are)17 b(constan)o(ts.)10 1710 y(Assume)e (inductiv)o(ely)f(that:)654 1789 y Fn(\015)682 1768 y Fi(\000)p Fj(c)725 1773 y Ff(3)742 1768 y Fl(\014)762 1773 y Ff(1)780 1768 y Fl(\025)801 1756 y Ff(2)817 1768 y Fl(k)853 1789 y Fn(<)f(Z)937 1796 y Fl(k)973 1789 y Fn(<)h(\015)1053 1768 y Fi(\000)p Fj(c)1096 1773 y Ff(4)1113 1768 y Fl(\014)1133 1773 y Ff(1)1150 1768 y Fl(\025)1171 1756 y Ff(2)1188 1768 y Fl(k)-13 1893 y Fp(for)i Fn(k)g Fm(\025)e Fn(h)p Fp(;)i(then)g(the)g(same)f(equation)h(holds)h(for)f Fn(Z)964 1900 y Fl(h)p Fi(\000)p Fj(1)1048 1893 y Fp(as,)h(from)e (eq.\(75\):)251 2050 y Fn(Z)284 2057 y Fl(h)p Fi(\000)p Fj(1)366 2050 y Fm(\024)f Fn(Z)452 2057 y Fl(h)474 2050 y Fp(\(1)e(+)f Fn(\014)606 2057 y Fj(1)625 2050 y Fn(c)646 2057 y Fj(4)666 2050 y Fn(\025)694 2029 y Fj(2)714 2050 y Fp(\))j Fm(\024)f Fn(\015)827 2029 y Fi(\000)p Fl(\014)874 2034 y Ff(1)892 2029 y Fl(c)907 2034 y Ff(4)924 2029 y Fl(\025)945 2018 y Ff(2)962 2029 y Fj(\()p Fl(h)p Fi(\000)p Fj(1\))1062 2016 y Fp(1)f(+)f Fn(\014)1175 2023 y Fj(1)1194 2016 y Fn(c)1215 2023 y Fj(4)1235 2016 y Fn(\025)1263 1998 y Fj(2)p 1062 2038 221 2 v 1106 2084 a Fn(e)1129 2070 y Fl(\014)1149 2075 y Ff(1)1166 2070 y Fl(c)1181 2075 y Ff(4)1199 2070 y Fl(\025)1220 2060 y Ff(2)1302 2050 y Fm(\024)i Fn(\015)1382 2029 y Fi(\000)p Fl(\014)1429 2034 y Ff(1)1447 2029 y Fl(c)1462 2034 y Ff(4)1479 2029 y Fl(\025)1500 2018 y Ff(2)1517 2029 y Fj(\()p Fl(h)p Fi(\000)p Fj(1\))227 2188 y Fn(Z)260 2195 y Fl(h)p Fi(\000)p Fj(1)342 2188 y Fm(\025)g Fn(Z)427 2195 y Fl(h)450 2188 y Fp(\(1)e(+)g(2)p Fn(\014)605 2195 y Fj(1)625 2188 y Fn(c)646 2195 y Fj(3)666 2188 y Fn(\025)694 2167 y Fj(2)714 2188 y Fp(\))j Fm(\025)f Fn(\015)827 2167 y Fi(\000)p Fl(\014)874 2172 y Ff(1)892 2167 y Fl(c)907 2172 y Ff(3)924 2167 y Fl(\025)945 2156 y Ff(2)962 2167 y Fj(\()p Fl(h)p Fi(\000)p Fj(1\))1062 2154 y Fp(1)f(+)f(2)p Fn(\014)1199 2161 y Fj(1)1218 2154 y Fn(c)1239 2161 y Fj(3)1259 2154 y Fn(\025)1287 2136 y Fj(2)p 1062 2176 245 2 v 1119 2222 a Fn(e)1142 2208 y Fl(\014)1162 2213 y Ff(1)1178 2208 y Fl(c)1193 2213 y Ff(3)1211 2208 y Fl(\025)1232 2198 y Ff(2)1326 2188 y Fm(\025)j Fn(\015)1407 2167 y Fi(\000)p Fl(\014)1454 2172 y Ff(1)1471 2167 y Fl(c)1486 2172 y Ff(3)1504 2167 y Fl(\025)1525 2156 y Ff(2)1542 2167 y Fj(\()p Fl(h)p Fi(\000)p Fj(1\))10 2292 y Fp(A)i(similar)e(argumen)o(t) i(can)g(b)q(e)g(used)h(to)f(treat)h(the)f(b)q(ound)h(on)g Fn(\033)1218 2299 y Fl(h)1256 2292 y Fp(so)g(that)g(eq.\(83\))f(is)g(v) o(eri\014ed.)10 2371 y(Eq.\(79\))24 b(follo)o(ws)f(assuming)g(that)h (eq.\(79\)\(83\)\(82\))g(hold)f(for)h Fn(k)j Fm(\025)f Fn(h)d Fp(so)h(that)g(there)e(are)i(t)o(w)o(o)-13 2449 y(constan)o(ts)17 b Fn(c)224 2456 y Fj(1)243 2449 y Fn(;)8 b(c)286 2456 y Fj(2)322 2449 y Fp(suc)o(h)16 b(that)h(from)e (eq.\(75\):)456 2576 y Fm(\000)p Fn(c)516 2583 y Fj(1)535 2576 y Fn(\025)563 2555 y Fj(2)584 2576 y Fp([)603 2542 y Fn(\033)631 2549 y Fl(h)p 603 2564 51 2 v 603 2610 a Fn(\015)631 2595 y Fl(h)669 2576 y Fp(+)c Fn(\015)746 2555 y Fl(h)768 2576 y Fp(])i Fn(<)h(\025)875 2583 y Fl(h)p Fi(\000)p Fj(1)954 2576 y Fm(\000)d Fn(\025)1032 2583 y Fl(h)1069 2576 y Fn(<)j(c)1142 2583 y Fj(2)1161 2576 y Fp([)1180 2542 y Fn(\033)1208 2549 y Fl(h)p 1180 2564 V 1180 2610 a Fn(\015)1208 2595 y Fl(h)1246 2576 y Fp(+)d Fn(\015)1323 2555 y Fl(h)1346 2576 y Fp(])p Fn(\025)1388 2555 y Fj(2)907 2728 y Fp(45)p eop %%Page: 46 46 46 45 bop -13 -5 a Fp(and)17 b(pro)q(ceeding)f(in)g(the)g(same)f(w)o(a) o(y)h(for)g Fn(\016)785 2 y Fl(h)p Fi(\000)p Fj(1)853 -5 y Fp(.)10 74 y(Finally)f(in)h(the)g(same)f(w)o(a)o(y)h(the)g (equations)h(from)e Fn(\034)977 81 y Fl(h)1000 74 y Fn(;)8 b(\022)1045 81 y Fl(h)1083 74 y Fp(are)16 b(obtained)h(from)e (eq.\(75\).)10 152 y Fg(aknow)r(le)n(dgments)g Fp(W)l(e)d(are)g(deeply) f(indebted)g(with)g(G.Benfatto)h(and)h(G.Galla)o(v)o(otti)d(for)j (suggesting)-13 230 y(us)j(the)g(sub)s(ject)g(of)h(this)f(w)o(ork)g (and)h(for)f(their)g(con)o(tin)o(uous)g(and)h(encouraging)g(advice.)-13 424 y Fo(6)83 b(References)-13 551 y Fd([A.)24 b Fp(])17 b(P)l(.W.)f(Anderson:)24 b Fg(Luttinger)19 b(liquid)h(b)n(ehaviour)e (of)h(cupr)n(ate)f(c)n(omp)n(ounds)f Fp(Ph)o(ysical)g(Review)109 630 y(Letters,)f(64,)g(1839-1845)k(\(1990\))-13 749 y Fd([B.G.)j Fp(])14 b(G.Benfatto)h(G.Galla)o(v)o(otti:)k Fg(Perturb)n(ation)e(the)n(ory)e(of)h(the)h(F)l(ermi)f(surfac)n(e)g(in) g(a)h(quantum)109 828 y(liquid.)36 b(A)21 b(gener)n(al)i(quasip)n (article)f(formalism)f(and)h(one)h(dimensional)f(systems)p Fp(,)g(Jour.)35 b(of)109 906 y(Stat.)21 b(Phis.,)16 b Fd(59)p Fp(,)g(541-664)j(\(1990\).)-13 1026 y Fd([B.G.1)k Fp(])16 b(G.Benfatto)g(G.Galla)o(v)o(otti:)k Fg(R)n(enormalization)d (gr)n(oup)p Fp(,)e(Princeton)h(Press)h(\(1995\).)-13 1145 y Fd([B.G.M)23 b Fp(])10 b(G.Benfatto)h(G.Galla)o(v)o (otti,V.Mastropietro:)16 b Fg(R)n(enormalization)c(gr)n(oup)g(and)h (the)g(F)l(ermi)109 1224 y(surfac)n(e)k(in)h(the)g(Luttinger)g(mo)n (del)p Fp(,Ph)o(ys.Rev.,)c Fd(B42)p Fp(,)h(5468-5480)20 b(\(1992\).)-13 1343 y Fd([B.G.P)-5 b(.S.)23 b Fp(])10 b(G.Benfatto,)h(G.Galla)o(v)o(otti,)g(A.Pro)q(cacci,)f(B.Scopp)q(ola:)p Fg(Beta)j(function)h(and)e(Schwinger)109 1422 y(functions)k(for)d(a)i (many)f(fermions)h(system)f(in)h(one)g(dimension.)22 b(A)o(nomaly)15 b(of)f(the)h(F)l(ermi)g(sur-)109 1500 y(fac)n(e)p Fp(,)h(Comm.Math.Ph)o(ys,)d Fd(160)p Fp(,)i(93-171)k (\(1994\).)-13 1620 y Fd([B.G.L.)j Fp(])j(G.Benfatto,)i(G.Galla)o(v)o (otti,)g(J.Leb)q(o)o(witz:)39 b Fg(Disor)n(der)25 b(in)h(the)g(1D)g (Holstein)i(mo)n(del)109 1698 y Fp(preprin)o(t)15 b(1994)-13 1818 y Fd([B.M.)23 b Fp(])18 b(F.Bonetto,)g(V.Mastropietro:)p Fg(R)n(enormalization)h(gr)n(oup)g(the)n(ory)g(in)h(a)g Fn(d)e Fp(=)g(1)i Fg(system)g(of)109 1896 y(inter)n(acting)e(fermions)g (in)g(a)f(p)n(erio)n(dic)f(p)n(otential)p Fp(,)h(CARR)f(preprin)o(t)f (25/93)-13 2016 y Fd([F.M.R.T.)22 b Fp(])f(J.)f(F)l(eldman,)g(J.)h (Magnen,)h(V.)e(Riv)m(asseau,)j(E.)e(T)l(rub)q(o)o(witz)g Fg(A)o(n)h(In\014nite)h(V)l(olume)109 2094 y(Exp)n(ansion)i(for)f(Many) h(F)l(ermion)g(Gr)n(e)n(en)-5 b('s)24 b(F)l(unction)p Fp(,)k(Helv.)44 b(Ph)o(ys.)h(Acta)24 b(65,679-721,)109 2172 y(\(1992\))-13 2292 y Fd([G.)f Fp(])18 b(G.)f(Galla)o(v)o(otti:)24 b Fg(R)n(enormalization)19 b(the)n(ory)f(and)h(the)g(ultr)n(aviolet)h (stability)g(for)e(sc)n(alar)g(\014elds)109 2370 y(via)f(r)n (enormalization)h(gr)n(oups)e(metho)n(ds)p Fp(,)g(Rev.Mo)q(d.Ph)o(ys.)k Fd(57)p Fp(,)c(471-562)j(\(1985\).)-13 2490 y Fd([K.)k Fp(])e(W.)h(Kohn:)32 b Fg(A)o(nalytic)23 b(pr)n(op)n(erties)f(of)g(Blo) n(ch)h(waves)g(and)g(Wannier)f(function)p Fp(,)i(Ph)o(ys.Rev.)109 2568 y Fd(115)p Fp(,)16 b(4,)g(809-821)j(\(1959\).)907 2728 y(46)p eop %%Page: 47 47 47 46 bop -13 -5 a Fd([L.)23 b Fp(])13 b(J.)h(Luttinger:)20 b Fg(A)o(n)c(exactly)g(soluble)h(mo)n(del)e(of)h(a)f(many)g(fermions)g (system)p Fp(,)f(Journal)g(of)g(Math-)109 74 y(ematical)g(Ph)o(ysics,)h Fd(4)p Fp(,)h(1154-1162)j(\(1963\).)-13 193 y Fd([L1.)k Fp(])15 b(J.)h(Luttinger:)21 b Fg(F)l(ermi)c(surfac)n(e)g(and)g(some)g (simple)h(e)n(quilibrium)g(pr)n(op)n(erties)e(of)h(a)g(system)g(of)109 272 y(inter)n(acting)h(fermions)p Fp(,)e(Ph)o(ysical)g(Review,)e(119,)j (1153-1163,)i(\(1960\).)-13 391 y Fd([Le.)k Fp(])17 b(A.)h(Lesniewski:) 24 b Fg(E\013e)n(ctive)d(action)f(for)e(the)i(Y)l(oukawa)g(2)f(quantum) i(\014eld)f(The)n(ory)p Fp(,)d(Comm.)109 470 y(Math.)k(Ph)o(ys.)g Fd(108)p Fp(,)16 b(437-467)j(\(1987\))-13 589 y Fd([L.E.)j Fp(])h(A.)f(Luther,)j(V.J.)d(Emery:)33 b Fg(Backwar)n(d)25 b(sc)n(attering)f(in)g(one)h(dimensional)g(ele)n(ctr)n(on)f(gas)109 668 y Fp(Ph)o(ys.)d(Rev.)f(Lett.,)c(53,10,)h(589-552,)h(\(1974\).)-13 787 y Fd([L.E.1)k Fp(])d(A.)g(Luther,)h(V.J.)e(Emery:)p Fg(Calculations)i(of)g(critic)n(al)h(exp)n(onents)h(in)e(2)h(dimension) g(fr)n(om)109 866 y(Quantum)d(Field)g(The)n(ory)e(in)i(one)g(dimension) f Fp(Ph)o(ys.)k(Rev.,)15 b(B12,)h(3908-3920)q(,)i(\(1975\).)-13 985 y Fd([L.W.)k Fp(])13 b(J.)f(Luttinger,)h(J.)f(W)l(ard:)p Fg(Gr)n(ound-state)j(ener)n(gy)f(of)g(a)g(many)g(fermion)h(system)f(II) p Fp(,)d(Ph)o(ysical)109 1064 y(Review,)j(118,)j(1417-1427)q(,)h (\(1960\).)-13 1183 y Fd([M.L)23 b Fp(])e(D.Mattis)h(E.Lieb:)31 b Fg(Exact)24 b(solution)f(of)f(a)h(many)f(fermion)h(system)f(and)h (its)f(asso)n(ciate)n(d)109 1262 y(b)n(oson)17 b(\014eld)p Fp(,)g(Journal)g(of)f(Mathematical)e(Ph)o(ysics)i Fd(6)p Fp(,)f(304)j(\(1965\).-state)f(energy)f(of)h(a)f(man)o(y)109 1340 y(fermion)e(system)h(I)q(I,)-13 1460 y Fd([S.)23 b Fp(])15 b(J.Soly)o(om:)k Fg(The)d(F)l(ermi)g(gas)h(mo)n(del)f(of)g (one)h(dimensional)h(c)n(onductors)p Fp(,)d(Adv)m(ance)g(in)f(Ph)o (ysics)109 1538 y Fd(28)p Fp(,)i(201-303)i(\(1958\))-13 1658 y Fd([T.P)-5 b(.)23 b Fp(])16 b(T)l(rub)q(o)o(witz)h(E.P)o(o)q (esc)o(hel)e(J.)h(:In)o(v)o(erse)e(sp)q(ectral)i(theory)l(,)g(Academic) d(Press,)j(1987)-13 1777 y Fd([W.F.)23 b Fp(])d(Wilson)g(K.,)h(Fisher)f (M.)f Fg(Critic)n(al)i(exp)n(onents)i(in)f(3.99)f(dimensions)p Fp(,)g(Ph)o(ys.Rev.)33 b(Lett.)109 1856 y(28,)16 b(240-243,)j(\(1972\)) 907 2728 y(47)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF